Bounded Variation and Around 9783110265118, 9783110265071

Table of contents :
Preface
Introduction
0 Prerequisites
0.1 The Lebesgue integral
0.2 Some functional analysis
0.3 Basic function spaces
0.4 Comments on Chapter 0
0.5 Exercises to Chapter 0
1 Classical BV-spaces
1.1 Functions of bounded variation
1.2 Bounded variation and continuity
1.3 Functions of bounded Wiener variation
1.4 Functions of several variables
1.5 Comments on Chapter 1
1.6 Exercises to Chapter 1
2 Nonclassical BV-spaces
2.1 The Wiener–Young variation
2.2 The Waterman variation
2.3 The Schramm variation
2.4 The Riesz–Medvedev variation
2.5 The Korenblum variation
2.6 Higher order Wiener-type variations
2.7 Comments on Chapter 2
2.8 Exercises to Chapter 2
3 Absolutely continuous functions
3.1 Continuity and absolute continuity
3.2 The Vitali–Banach–Zaretskij theorem
3.3 Reconstructing a function from its derivative
3.4 Rectifiable functions
3.5 The Riesz–Medvedev theorem
3.6 Higher order Riesz-type variations
3.7 Comments on Chapter 3
3.8 Exercises to Chapter 3
4 Riemann–Stieltjes integrals
4.1 Classical RS-integrals
4.2 Bounded variation and duality
4.3 Bounded p-variation and duality
4.4 Nonclassical RS-integrals
4.5 Comments on Chapter 4
4.6 Exercises to Chapter 4
5 Nonlinear composition operators
5.1 The composition operator problem
5.2 Boundedness and continuity
5.3 Spaces of differentiable functions
5.4 Global Lipschitz continuity
5.5 Local Lipschitz continuity
5.6 Comments on Chapter 5
5.7 Exercises to Chapter 5
6 Nonlinear superposition operators
6.1 Boundedness and continuity
6.2 Lipschitz continuity
6.3 Uniform boundedness and continuity
6.4 Functions of several variables
6.5 Comments on Chapter 6
6.6 Exercises to Chapter 6
7 Some applications
7.1 Convergence criteria for Fourier series
7.2 Fourier series and Waterman spaces
7.3 Applications to nonlinear integral equations
7.4 Comments on Chapter 7
References
List of functions
List of symbols
Index

Citation preview

Jürgen Appell, Józef Banaś, Nelson Merentes Bounded Variation and Around

De Gruyter Studies in Nonlinear Analysis and Applications

| Editor in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Karl-Heinz Hoffmann, Munich, Germany Mikio Kato, Nagano, Japan Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Boris N. Sadovsky, Voronezh, Russia Alfonso Vignoli, Rome, Italy Katrin Wendland, Freiburg, Germany

Volume 17

Jürgen Appell, Józef Banaś, Nelson Merentes

Bounded Variation and Around |

Mathematics Subject Classification 2010 Primary: 26A45, 47H30; Secondary: 26A15, 26A16, 26A30, 26A42, 26A46, 45G05, 45G10, 46B25 Authors Prof. Dr. Jürgen Appell University of Würzburg Department of Mathematics Emil-Fischer-Str. 30 97074 Würzburg Germany [email protected] Prof. Dr. Józef Banaś Rzeszów University of Technology Department of Mathematics al. Powstańców Warszawy 8 35-959 Rzeszów Poland [email protected] Prof. Dr. Nelson Merentes Central University of Venezuela School of Mathematics Paseo Los Ilustres Caracas 1020 Venezuela [email protected]

ISBN 978-3-11-026507-1 e-ISBN 978-3-11-026511-8 Set-ISBN 978-3-11-026624-5 ISSN 0941-813X Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2014 Walter de Gruyter GmbH, Berlin/Boston Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ♾Printed on acid-free paper Printed in Germany www.degruyter.com

Preface Functions of bounded variation are fascinating mathematical objects which, since their definition by C. Jordan more than a century ago, have found much attention in real analysis, functional analysis, measure theory, integration theory, operator theory, Fourier analysis, nonlinear analysis, and even some fields of mathemati­ cal physics. Also, Jordan’s original definition has been extended to more general classes of functions which are often motivated by (and in fact, lead to) convergence criteria for Fourier series or existence results for Riemann–Stieltjes integrals. Such general classes are of independent interest in the study of dual spaces in linear func­ tional analysis and operator theory. For example, the classical Riemann–Stieltjes integral provides a natural one-to-one correspondence between the dual of the space C([𝑎, 𝑏]) of continuous functions and the space NBV([𝑎, 𝑏]) of (normalized) functions of bounded variation. The basic facts about functions of bounded variation may be found in virtually any textbook on real or functional analysis. Among the application-oriented books which use functions of bounded variation, we mention the classical work of E. Giusti [122] and the recent monograph by L. Ambrosio, N. Fusco and D. Pallara [7]. However, to the best of our knowledge, there are no books which provide a thorough and self-­ contained account of functions of (generalized) bounded variation as an independent subject, the methods connected with their study, nor their applications to various an­ alytical or geometrical problems. An exception is the French Lecture Notes Variation totale d’une fonction by M. Bruneau (1974, see [65] in the list of references) which, how­ ever, covers topics other than those treated here. Some kind of “rudimentary prede­ cessor” of this book is the Spanish monograph El operador de composición en espa­ cios de funciones con algún tipo de variación acotada by the third author and S. Rivas (1996, [226]). We remark that the monograph [226] does not contain full proofs, and only sketches very few applications to composition operators. Roughly speaking, the whole material treated in [226] is contained in Chapters 1, 2, 5, and 6 of the present book. The purpose of this monograph is four-fold. Firstly, we want to collect the basic facts about functions of bounded variation and related functions, such as Lipschitz continuous or absolutely continuous functions, to present the main ideas which have been shown to be useful in studying their properties, and to provide a comparison of their importance and suitability for applications. Secondly, we want to study the (sometimes quite surprising and pathological) behavior of nonlinear composition and superposition operators in spaces of functions of bounded variation with a particular emphasis on continuity and boundedness in norm, or global and local Lipschitz con­ ditions. (It is the second part of this book, treating nonlinear operator theory, which may be considered as complementary to the standard reference [271] and motivated us to submit it to the De Gruyter Series in Nonlinear Analysis and Applications.)

vi | Preface Third, some topics like Riemann–Stieltjes integrals and their role in the duality theory of Banach spaces which have not been considered in the above mentioned Spanish monograph will be treated in some detail. Finally, we will also discuss some newer developments which were still unknown when the monograph [226] appeared in 1996. In the last sections of nearly every chapter called “Comments,” we discuss ad­ ditional or peripheral topics of interest, alternate presentations, and historical com­ mentary. Moreover, we try to take into account recent results and state several open problems; therefore, this book might also be a fruitful source of inspiration for young researchers and postgraduate students who are interested in the subject. The reader will notice that apart from recent developments and open problems, the last section of each chapter contains a considerable number of exercises. This may be somewhat unusual for a research monograph; however, we are convinced that such exercises provide a deeper insight into the topic and might be useful for students dur­ ing lectures and seminars on higher analysis. For a variety of reasons, we have deemed these exercises important for a full understanding of the material. Some of them in­ clude straightforward “computations,” some are simply detail checking, and some un­ veil the seeds of ideas essential for later developments. Exercises which, in our opin­ ion, are more technical are marked with an *, but this is sometimes a matter of taste. Regarding the book as a whole, the choice of topics is necessarily far from compre­ hensive and has been made with a number of criteria in mind. The most obvious one is mathematical importance, but some additional topics were chosen because it is pos­ sible to discuss them in an easily accessible way, others because they have some un­ usual feature, and some because the authors felt that certain results should be treated separately in the comment section so as not to overburden the presentation in the main sections. The only prerequisite for understanding this book is a modest background in real analysis, functional analysis, and operator theory. It addresses nonspecialists who want to get an idea of the development of the theory, methods, and applications of different notions of functions of bounded variation in the last 50 years as well as a glimpse into the diversity of the directions in which current research is moving. Apart from abstract results, we have tried to present many examples which give some indi­ cation as to the flavor of the subject. There are now several good books available with a particular emphasis on examples; we only mention the classical work Counterexam­ ples in Analysis by B. R. Gelbaum and J. M. H. Olmstedt (1964, see [118]) and the more recent Russian book Real Analysis in Exercises by A. N. Bakhvalov et al. (2005, [39]) which is an almost inexhaustible source of beautiful examples and counterexamples. The large number of examples is explained by the authors’ conviction that mono­ graphs of this kind would not be much use if all they gave were formal definitions, abstract theorems, and technical proofs. To understand a concept, one needs to know what it means intuitively, why it is important, and why it was first introduced. Above all, if it is a fairly general concept, then one wants to know some good examples – ones that are not too simple nor too complicated. In fact, a good example is much

Preface | vii

easier to understand than a general definition, and more experienced readers will be able to work out a general definition by “abstracting” the important properties from an example, rather than finding a significant example by “concretizing” a general defini­ tion. Or, to put it the way one of our colleagues recently said: “To understand a theory, you needn’t know all the theorems, but you have to know all the relevant examples!” Some parts of the presentation may seem redundant, but we don’t think that this is a flaw. For example, the spaces WBV𝑝 ([𝑎, 𝑏]) and RBV𝑝 ([𝑎, 𝑏]) of functions of bounded 𝑝-variation in the sense of Wiener and Riesz, respectively, are special cases of the more general spaces 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) and 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) of functions of bounded 𝜙-variation in the sense of Young and Medvedev, respectively, which may be obtained by the special choice 𝜙(𝑢) = |𝑢|𝑝 . However, instead of treating the general BV𝜙 -spaces right from the beginning, we start with the more special BV𝑝 -spaces to make the concepts and results more transparent. Similarly, instead of presenting a fairly general characterization of locally Lipschitz continuous composition operators between the spaces Lip([𝑎, 𝑏]) and BV([𝑎, 𝑏]) (Theorem 5.10), we first prove the same characterization for operators from BV([𝑎, 𝑏]) into itself (Theorem 5.9), which has been not only the historical start­ ing point of such results, but also the standard model of subsequent techniques for proving them. As for the order of the chapters themselves, the aim has been to make it the most natural one from a pedagogical point of view and to give the book some sense of direction. The notation used in this book is standard. A detailed symbol index at the end may be helpful to find special notations, such as the numerous variations we consider in Chapters 1 and 2. The end of a definition is marked by ◼, the end of a proof by ◻, and the end of an example by ♥. This book could not have been realized without the possibility of meetings in Germany, Poland, and Venezuela, generously supported by the European Commis­ sion and the Ministry of Education and Research of Venezuela. In particular, the first author gratefully acknowledges hospitality of the Central University of Cara­ cas (Venezuela) and of the Rzeszów University of Technology (Poland). A large debt of gratitude is also owed to our colleagues Daria Bugajewska, Dariusz Bugajewski, Janusz Matkowski, and Martin Väth for looking through some parts of an earlier ver­ sion of the manuscript. Of course, the reader has to thank them for each passage that “works,” and only blame us for those (hopefully, not too numerous) that don’t. Last, but by no means least, it is a great pleasure to thank Friederike Dittberner, Silke Hutt, and Anja Möbius from De Gruyter-Verlag Berlin, as well as le-tex Leipzig, for their a.i. (almost infinite) patience and constant kind support. Würzburg, Rzeszów, Caracas, Summer 2013

The Authors

Contents Preface | v Introduction | 1 0 0.1 0.2 0.3 0.4 0.5

Prerequisites | 7 The Lebesgue integral | 7 Some functional analysis | 18 Basic function spaces | 25 Comments on Chapter 0 | 43 Exercises to Chapter 0 | 46

1 1.1 1.2 1.3 1.4 1.5 1.6

Classical BV-spaces | 55 Functions of bounded variation | 55 Bounded variation and continuity | 71 Functions of bounded Wiener variation | 84 Functions of several variables | 91 Comments on Chapter 1 | 100 Exercises to Chapter 1 | 104

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Nonclassical BV-spaces | 112 The Wiener–Young variation | 112 The Waterman variation | 125 The Schramm variation | 152 The Riesz–Medvedev variation | 161 The Korenblum variation | 169 Higher order Wiener-type variations | 182 Comments on Chapter 2 | 187 Exercises to Chapter 2 | 202

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Absolutely continuous functions | 208 Continuity and absolute continuity | 208 The Vitali–Banach–Zaretskij theorem | 211 Reconstructing a function from its derivative | 218 Rectifiable functions | 231 The Riesz–Medvedev theorem | 240 Higher order Riesz-type variations | 244 Comments on Chapter 3 | 249 Exercises to Chapter 3 | 260

x | Contents 4 4.1 4.2 4.3 4.4 4.5 4.6

Riemann–Stieltjes integrals | 268 Classical RS-integrals | 268 Bounded variation and duality | 292 Bounded 𝑝-variation and duality | 298 Nonclassical RS-integrals | 302 Comments on Chapter 4 | 311 Exercises to Chapter 4 | 316

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Nonlinear composition operators | 324 The composition operator problem | 324 Boundedness and continuity | 344 Spaces of differentiable functions | 354 Global Lipschitz continuity | 364 Local Lipschitz continuity | 368 Comments on Chapter 5 | 377 Exercises to Chapter 5 | 382

6 6.1 6.2 6.3 6.4 6.5 6.6

Nonlinear superposition operators | 385 Boundedness and continuity | 385 Lipschitz continuity | 400 Uniform boundedness and continuity | 406 Functions of several variables | 415 Comments on Chapter 6 | 419 Exercises to Chapter 6 | 423

7 7.1 7.2 7.3 7.4

Some applications | 425 Convergence criteria for Fourier series | 425 Fourier series and Waterman spaces | 429 Applications to nonlinear integral equations | 435 Comments on Chapter 7 | 444

References | 453 List of functions | 467 List of symbols | 468 Index | 472

Introduction In 1829, Johan Peter Gustav Lejeune Dirichlet proved that the Fourier series of a piece­ wise monotone real function is pointwise convergent. This result is now known as the Dirichlet criterion in the theory of Fourier series and seems to be the first mathemati­ cally rigorous proof of Fourier’s conjecture which was raised in 1807 and published in his pioneering work Théorie analytique de la chaleur on the representability of func­ tions by means of a trigonometric series. According to Béla Szökefalvi–Nagy, the his­ tory of the Fourier series started with an exchange of letters between Jean–Baptiste d’Alembert, Leonhard Euler, and Daniel Bernoulli on the problem of the clamped vi­ brating string. A further milestone was Camille Jordan’s paper Sur les séries de Fourier (1881) in which he introduced the notion of functions of bounded variation and extended the Dirichlet criterion to this class of functions. In the same paper, he also proved that a function has bounded variation if and only if it may be represented as a difference of two monotonically increasing functions. In modern terminology, this means that the space of functions of bounded variation on [𝑎, 𝑏], usually denoted by BV([𝑎, 𝑏]), is the linear hull of the set of all monotone functions (which do not form a linear space on their own). Subsequently, Jordan’s definition has been generalized and extended in various directions, both from the viewpoint of the theory of real functions and from more application-oriented viewpoints. For example, in 1915, Charles De la Vallée–Poussin introduced the class of functions of bounded second variation and proved that every such function can be represented as a difference of two convex functions. In 1934, Mi­ hael T. Popoviciu proved a parallel result for higher order variations, building on the notion of functions of bounded 𝑘-th variation. In case 𝑘 = 1, one recovers Jordan’s de­ composition of classical BV-functions and in case 𝑘 = 2, the De la Vallée–Poussin decomposition. While these results seem to be more of theoretical interest, two other extensions have turned out to be extremely useful in applications to Fourier series, namely, Nor­ bert Wiener’s definition of functions of bounded 𝑝-variation (1924) and, more gener­ ally, Laurence C. Young’s definition of functions of bounded 𝜙-variation (1937), where 𝜙 is a suitable convex increasing “gauge function.” A different notion of functions of bounded 𝑝-variation has been introduced by Frigyes Riesz (1910), and its generaliza­ tion to bounded 𝜙-variation by Yurij T. Medvedev (1953). Loosely speaking, it can be said that passing from 𝑝-variations to 𝜙-variations is similar to passing from Lebesgue spaces to Orlicz spaces. The variations introduced by Riesz and Medvedev seem to be very natural from the functional analytic point of view. In fact, an important re­ sult (called the Riesz lemma by some people) states that a function 𝑓 has bounded 𝑝-variation in Riesz’s sense (1 < 𝑝 < ∞) if and only if 𝑓 is absolutely continuous and its derivative 𝑓󸀠 exists almost everywhere and belongs to the Lebesgue space L𝑝 .

2 | Introduction (This means that Riesz introduced Sobolev spaces, at least in the scalar case, several years prior to Sobolev.) A parallel result holds for functions 𝑓 of bounded 𝜙-variation in Medvedev’s sense, where the L𝑝 -condition on the derivative 𝑓󸀠 has to be replaced by some appropriate integrability condition for 𝜙 ∘ 𝑓󸀠 . Clearly, an analogous result for 𝑝 = 1 is not true because functions in RBV1 = BV are generally not continuous, let alone absolutely continuous. We notice that a “higher order variant” of Riesz’s characterization leads to the following question: how can we characterize, by means of bounded variation tech­ niques, all absolutely continuous functions 𝑓 whose second derivative 𝑓󸀠󸀠 exists al­ most everywhere and belongs to L𝑝 ? This problem was solved in 1991 by the third au­ thor who introduced the class of functions of bounded (𝑝, 2)-variation, providing in this way a unified approach to the Riesz theory and the De la Vallée–Poussin theory. In the meantime, yet more general notions of bounded variation have been introduced and studied by Daniel Waterman (1976), Michael Schramm (1982), Boris Korenblum (1975), Hwa Jun Kim (2006), and others. All of these notions and results, together with a large variety of applications, are scattered over many research papers which appeared during the last 50 years and are contained in the detailed list of references at the end of this book. However, as far as we know, they have not been collected in a single monograph, and thus the aim of this book is to fill the gap. As far as applications are concerned, let us mention four fields in which functions of bounded variation turn out to be useful: – They admit a decomposition into, hopefully, simpler functions. – They are connected to geometric notions like curve length or surface area. – They make it possible to define Riemann–Stieltjes type integrands. – They provide (uniform) convergence results for Fourier series. For the classical Jordan space BV([𝑎, 𝑏]), this program has been accomplished quite successfully over the past 100 years: – BV-functions may be represented as differences of increasing functions. – BV-functions are precisely those whose graphs are rectifiable curves. – BV-functions define Riemann–Stieltjes integrals for continuous integrands. – BV-functions have, after a suitable normalization, convergent Fourier series. As we will see, for the other spaces of functions of (generalized) variation, this ambi­ tious program can be fulfilled only in part. This monograph consists of 8 chapters. In Chapter 0, which is introductory, we col­ lect everything on measure and integration, function spaces, and functional analysis which will be needed in the following chapters. Chapter 1 starts with the definition and discussion of the classical space BV([𝑎, 𝑏]) of real functions of bounded variation on [𝑎, 𝑏], including Jordan’s decomposition of functions in BV([𝑎, 𝑏]) as differences of two increasing functions on [𝑎, 𝑏]. We also consider the spaces WBV𝑝 ([𝑎, 𝑏]) (1 ≤ 𝑝 < ∞) of

Introduction

| 3

functions of bounded 𝑝-variation (in Wiener’s sense) and discuss their basic properties and connections with other function spaces, with a particular emphasis on spaces of continuous, Lipschitz continuous, Hölder continuous, and absolutely continuous func­ tions. In this chapter, we will also briefly discuss functions of several (in particular, two) variables, mainly due to Vyacheslav V. Chistyakov. The corresponding results are rather technical, but seem to have important applications. As mentioned before, the space BV([𝑎, 𝑏]) has been generalized in various direc­ tions. Norbert Wiener and Laurence C. Young distorted the measurement of intervals in the range of functions by considering 𝑝-th powers or, more generally, continuous increasing gauge functions 𝜙, introducing the classes WBV𝑝 ([𝑎, 𝑏]) and WBV𝜙 ([𝑎, 𝑏]). Subsequently, Daniel Waterman and Michael Schramm admitted countable families of such gauge functions in order to generalize the concept of variation, which leads to the classes Λ BV([𝑎, 𝑏]) and 𝛷 BV([𝑎, 𝑏]). As mentioned before, however, the most interesting generalization has been introduced by Frigyes Riesz and extended by Yurij T. Medvedev to the setting of gauge functions; we denote the corresponding classes by RBV𝑝 ([𝑎, 𝑏]) and RBV𝜙 ([𝑎, 𝑏]), respectively. All of these generalizations of the con­ cept of bounded variation will be discussed in Chapter 2 which is, by far, the largest chapter in the book. A flaw of all the extensions described above is the loss of an effective decomposi­ tion of a function from the corresponding classes into, hopefully, simpler functions, such as for Jordan’s classical space BV([𝑎, 𝑏]). In 1975, Boris Korenblum considered a new kind of variation, called 𝜅-variation, introducing a function 𝜅 for distorting the length |𝑡𝑗 − 𝑡𝑗−1 | of the subinterval [𝑡𝑗−1 , 𝑡𝑗 ] in a partition {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } of the underly­ ing interval, rather than the expression |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| in the range. One advantage of this alternate approach is that a function of bounded 𝜅-variation may be decomposed into the difference of two simpler functions called 𝜅-decreasing functions (the precise definition will be given in Section 2.5 below). Absolutely continuous functions are intimately related to functions of bounded variation in several respects. First of all, absolute continuity is equivalent to the com­ bination of three properties, namely, continuity, bounded variation, and invariance of (Lebesgue) nullsets; this is the assertion of the famous Vitali–Banach–Zaretskij theo­ rem. Second, as mentioned above, the absolutely continuous functions on [𝑎, 𝑏] are precisely those functions 𝑓 ∈ BV([𝑎, 𝑏]) whose derivatives 𝑓󸀠 (which exist almost ev­ erywhere, by Lebesgue’s differentiation theorem) belong to L1 ([𝑎, 𝑏]); moreover, the fundamental theorem of calculus (in the Lebesgue integral version) holds in this case which makes it possible to recover an absolutely continuous function, up to an addi­ tive constant, from its derivative. If we replace the condition 𝑓 ∈ AC([𝑎, 𝑏]) with the stronger condition 𝑓 ∈ RBV𝑝 ([𝑎, 𝑏]) (1 < 𝑝 < ∞) in this statement, we get precisely the condition 𝑓󸀠 ∈ L𝑝 ([𝑎, 𝑏]); this will be discussed in further detail, together with some generalization due to Yurij T. Medvedev, in Chapter 3. In Chapter 4, we study Riemann–Stieltjes integrals. It is well known that by means of the classical Riemann–Stieltjes integral, we can construct a surjective isometry

4 | Introduction between the dual space of the Chebyshev space 𝐶([𝑎, 𝑏]) and a subspace of (suit­ ably normalized) functions in BV([𝑎, 𝑏]). Similarly, an analogous isometry makes it possible to identify the dual space of the Lebesgue space L𝑝 ([𝑎, 𝑏]) (1 < 𝑝 < ∞) with the space RBV𝑝/(𝑝−1) ([𝑎, 𝑏]), where RBV𝑝 ([𝑎, 𝑏]) denotes the space of functions of bounded 𝑝-variation in Riesz’s sense introduced in Chapter 2. This one-to-one corre­ spondence is different from the usual identification of the dual space of L𝑝 ([𝑎, 𝑏]) with L𝑝/(𝑝−1) ([𝑎, 𝑏]); moreover, it does not require regularization here since all functions of bounded Riesz 𝑝-variation are continuous for 𝑝 > 1. We may also consider the same isometry for the case 𝑝 = 1 which leads to the space RBV∞ ([𝑎, 𝑏]) = Lip([𝑎, 𝑏]) of Lipschitz continuous functions on [𝑎, 𝑏], but not for the case 𝑝 = ∞, which requires a different approach. Given a function ℎ : ℝ → ℝ, in Chapter 5, we study the (autonomous) nonlinear composition operator Cℎ defined by Cℎ 𝑓(𝑥) := ℎ(𝑓(𝑥))

(𝑎 ≤ 𝑥 ≤ 𝑏)

(1)

for 𝑓 : [𝑎, 𝑏] → ℝ belonging to several spaces of functions of (generalized) bounded variation and related spaces. It turns out that a typical condition, both necessary and sufficient, for Cℎ to map such a space into itself is a local Lipschitz condition on ℎ on the real line, i.e. |ℎ(𝑢) − ℎ(𝑣)| ≤ 𝑘(𝑟)|𝑢 − 𝑣| for |𝑢|, |𝑣| ≤ 𝑟. Afterwards, we will briefly con­ sider sufficient conditions on ℎ under which the corresponding operator C ℎ is bounded and/or continuous in norm, or even uniformly continuous on bounded sets or on the whole space. In view of the applications, Lipschitz conditions for the operator Cℎ are of particular interest. In the second part of Chapter 5, we will show that a global Lips­ chitz condition for Cℎ in norm often leads to a strong degeneracy for ℎ, namely, to affine functions of the form ℎ(𝑢) = 𝛼 + 𝛽𝑢. Roughly speaking, this means that whenever a global Lipschitz condition is imposed on the operator Cℎ , the underlying problem is necessarily linear, and thus of very limited interest. On the other hand, a local Lips­ chitz condition for Cℎ is fulfilled for sufficiently large classes of nonlinear functions ℎ (typically, those whose derivative ℎ󸀠 exists and is locally Lipschitz). This emphasizes the need of imposing local Lipschitz conditions, rather than global conditions, if we want to apply classical fixed point principles like Banach’s contraction mapping theo­ rem. In Chapter 6, we try to develop a parallel theory for the (nonautonomous) nonlin­ ear superposition operator Sℎ defined by Sℎ 𝑓(𝑥) = ℎ(𝑥, 𝑓(𝑥))

(𝑎 ≤ 𝑥 ≤ 𝑏) ,

(2)

where ℎ is now defined on the product [𝑎, 𝑏] × ℝ. It turns out that many results which hold in the autonomous case ℎ : ℝ → ℝ are not true in the nonautonomous case, or simply unknown. Loosely speaking, the “terra incognita” is much larger here, and many problems are open. For example, conditions which are both necessary and suffi­ cient for the function ℎ to guarantee that the corresponding operator (2) maps a func­ tion space into itself are known only in a few exceptional cases.

Introduction

| 5

Here is an example of what we mean. It is not hard to show that the autonomous operator (1) maps the space 𝐶([𝑎, 𝑏]) into itself (i.e. 𝑥 󳨃→ ℎ(𝑓(𝑥)) is continuous when­ ever 𝑥 󳨃→ 𝑓(𝑥) is continuous) if and only if ℎ is continuous on the real line. Similarly, the operator (1) maps the space 𝐶1 ([𝑎, 𝑏]) into itself (i.e. 𝑥 󳨃→ ℎ(𝑓(𝑥)) is continuously differentiable whenever 𝑥 󳨃→ 𝑓(𝑥) is continuously differentiable) if and only if ℎ is con­ tinuously differentiable on the real line. The first result carries over without change to the nonautonomous case: the operator (2) maps the space 𝐶([𝑎, 𝑏]) into itself (i.e. 𝑥 󳨃→ ℎ(𝑥, 𝑓(𝑥)) is continuous whenever 𝑥 󳨃→ 𝑓(𝑥) is continuous) if and only if ℎ is continuous on [𝑎, 𝑏] × ℝ. In contrast to this, however, the operator (2) may map the space 𝐶1 ([𝑎, 𝑏]) into itself (i.e. 𝑥 󳨃→ ℎ(𝑥, 𝑓(𝑥)) is continuously differentiable when­ ever 𝑥 󳨃→ 𝑓(𝑥) is continuously differentiable) even if ℎ is discontinuous somewhere on [𝑎, 𝑏] × ℝ! A corresponding sophisticated counterexample, due to Janusz Matkowski, is given in Section 6.2. The final Chapter 7 is concerned with a few selected applications. Historically, the most important applications of the space BV and its various generalizations refer to the Fourier series. In the first two sections of Chapter 7, we will outline some of these applications, with a particular emphasis on the Waterman space Λ BV. Building on the existence and uniqueness results for nonlinear problems obtained in Chapters 5 and 6, we will also consider applications to integral equations. We remark, however, that the main purpose of this monograph is to outline the theory, so little attention is given to the applications themselves. Needless to say, the few applications discussed in the last chapter are by no means exhaustive; we hope that the material contained in this book will be a source for further research. As pointed out in the Preface, we consider examples, counterexamples, and open problems of fundamental importance to get a deeper insight into the subject. Surpris­ ingly enough, some of these open problems are sometimes easily formulated, but ap­ parently hard to solve. Here are two typical examples. The first question is related to admissible “inner functions” (i.e. changes of variables) which preserve bounded variation: – Give conditions on a map 𝜏 : [𝑎, 𝑏] → [𝑎, 𝑏], possibly both necessary and sufficient, such that 𝑓 ∘ 𝜏 ∈ BV([𝑎, 𝑏]) for all 𝑓 ∈ BV([𝑎, 𝑏]). It is easy to see that monotonicity of 𝜏 is sufficient, but not necessary. Less obvious is the fact that bounded variation of 𝜏 is not sufficient. As we will see in Proposition 1.17, the correct condition on 𝜏 is pseudomonotonicity, a property which is intermediate between monotonicity and bounded variation. The second question is related to admissible “outer functions” which preserve bounded variation and may be formulated as follows. Let the space BV([𝑎, 𝑏]) be equipped with the natural norm ‖𝑓‖𝐵𝑉 = |𝑓(𝑎)| + Var (𝑓; [𝑎, 𝑏]) ,

(3)

6 | Introduction where Var (𝑓; [𝑎, 𝑏]) denotes the total variation of 𝑓 on [𝑎, 𝑏]. Now, given ℎ : ℝ → ℝ, consider the (autonomous) composition operator Cℎ defined by (1), i.e. Cℎ 𝑓 := ℎ ∘ 𝑓. Then the following three natural problems arise: – Give conditions on ℎ, possibly both necessary and sufficient, under which ℎ ∘ 𝑓 ∈ BV([𝑎, 𝑏]) for all 𝑓 ∈ BV([𝑎, 𝑏]), which means that the operator Cℎ maps the space BV into itself. – Give conditions on ℎ, possibly both necessary and sufficient, under which the op­ erator Cℎ is bounded in the norm (3). – Give conditions on ℎ, possibly both necessary and sufficient, under which the op­ erator Cℎ is continuous in the norm (3). The first problem was solved by Michael Josephy in 1981 who showed that Cℎ maps BV into itself if and only if ℎ is locally Lipschitz on the real line. The answer to the second question is almost trivial: a straightforward calculation shows that the operator Cℎ is always bounded in the norm (3) whenever it maps BV into itself. On the other hand, the answer to the third question is unknown: one does not know necessary and sufficient conditions for the continuity of the operator Cℎ in the norm (3). (Of course, conditions which are just sufficient are easily found.) In particular, we do not know whether con­ tinuity of Cℎ in norm follows from local Lipschitz continuity of ℎ on ℝ, and we believe that a counterexample, if there is any, will presumably be rather complicated. Even worse, if we replace the autonomous operator (1) by the nonautonomous operator (2), all three questions stated above are open.

0 Prerequisites This chapter is introductory and collects some of the material which will be needed in subsequent chapters. First, we recall basic notions and facts about (Lebesgue) in­ tegrable functions, with a particular emphasis on L𝑝 spaces. Afterwards, we discuss some concepts from functional analysis, including the Hahn–Banach theorem and its consequences, and introduce several function spaces which are related to functions of bounded variation. Since some of our results are standard and beyond the scope of this monograph, we state them without proofs. The proofs of such results may be found in any textbook on measure theory or functional analysis. We remark that, al­ though we restrict ourselves to subsets of the real line, many results carry over without change to measurable subsets of the Euclidean space ℝ𝑛 .

0.1 The Lebesgue integral We assume that the reader is familiar with the construction of the Lebesgue measure and Lebesgue integral on the real line. We will denote the Lebesgue measure of a mea­ surable set 𝑀 ⊆ ℝ by 𝜆(𝑀). As usual, properties are supposed to hold a.e. (almost everywhere), i.e. outside a Lebesgue nullset. We start with a result which shows that convergence a.e. of a sequence of measur­ able functions means, roughly speaking, uniform convergence up to a set of arbitrarily small measure. Theorem 0.1 (Egorov). Let 𝑀 ⊆ ℝ be a measurable set with 𝜆(𝑀) < ∞, and let (𝑓𝑛 )𝑛 be a sequence of measurable functions 𝑓𝑛 : 𝑀 → ℝ satisfying 𝑓𝑛 (𝑥) → 𝑓(𝑥)

(𝑛 → ∞)

(0.1)

a.e. on 𝑀. Then for each 𝜀 > 0, there exists a measurable subset M𝜀 ⊆ 𝑀 such that 𝜆(𝑀 \ M𝜀 ) < 𝜀 and (𝑓𝑛 )𝑛 converges uniformly on M𝜀 to 𝑓. The following result shows that measurability of a function means continuity up to a set of arbitrarily small measure. Theorem 0.2 (Luzin). Let 𝑀 ⊆ ℝ be a measurable set and 𝑓 : 𝑀 → ℝ a function. Then 𝑓 is measurable on 𝑀 if and only if for each 𝜀 > 0, we can find a closed subset M𝜀 ⊆ 𝑀 such that 𝜆(𝑀 \ M𝜀 ) < 𝜀 and the restriction 𝑓|M𝜀 : M𝜀 → ℝ of 𝑓 to the set M𝜀 is continuous. The following Theorems 0.3–0.6 refer to sequences of Lebesgue integrable functions; the proofs may be found in any textbook on measure theory.

8 | 0 Prerequisites Theorem 0.3 (Levi). Let 𝑀 ⊆ ℝ be a measurable set, and let (𝑓𝑛 )𝑛 be an increasing sequence of nonnegative measurable functions 𝑓𝑛 : 𝑀 → ℝ satisfying (0.1) everywhere on 𝑀. Then (0.2) ∫ 𝑓(𝑥) 𝑑𝑥 = lim ∫ 𝑓𝑛 (𝑥) 𝑑𝑥 , 𝑛→∞

𝑀

𝑀

where the functions and integrals are allowed to be infinite. Theorem 0.4 (Lebesgue). Let 𝑀 ⊆ ℝ be a measurable set, and let (𝑓𝑛 )𝑛 be a sequence of measurable functions 𝑓𝑛 : 𝑀 → ℝ satisfying (0.1) a.e. on 𝑀. Suppose that there exists an integrable function 𝐹 : 𝑀 → ℝ such that |𝑓𝑛 (𝑥)| ≤ 𝐹(𝑥)

(𝑛 ∈ ℕ, 𝑥 ∈ 𝑀) .

(0.3)

Then 𝑓 is also integrable and satisfies (0.2). Theorem 0.5 (Fatou). Let 𝑀 ⊆ ℝ be a measurable set, and let (𝑓𝑛 )𝑛 be a sequence of nonnegative measurable functions 𝑓𝑛 : 𝑀 → ℝ satisfying (0.1) a.e. on 𝑀. Then ∫ 𝑓(𝑥) 𝑑𝑥 ≤ lim inf ∫ 𝑓𝑛 (𝑥) 𝑑𝑥 . 𝑛→∞

𝑀

(0.4)

𝑀

Theorem 0.6 (absolute continuity of the integral). Let 𝑀 ⊆ ℝ be a measurable set, and let 𝑓 : 𝑀 → ℝ be integrable. Then for each 𝜀 > 0, there exists a 𝛿 > 0 such that for any measurable subset 𝑁 ⊆ 𝑀 with 𝜆(𝑁) ≤ 𝛿, we have ∫ |𝑓(𝑥)| 𝑑𝑥 ≤ 𝜀 .

(0.5)

𝑁

The following result refers to measurable functions defined on the Euclidean space ℝ𝑛 and gives a sufficient condition under which the “order of integration” on lower dimensional sections may be interchanged. Theorem 0.7 (Fubini). Suppose that 𝑓 : ℝ𝑝 ×ℝ𝑞 → ℝ is integrable on ℝ𝑛 , where 𝑝+𝑞 = 𝑛 and 𝑓 is written in the form 𝑧 = 𝑓(𝑥, 𝑦) (𝑥 ∈ ℝ𝑝 , 𝑦 ∈ ℝ𝑞 ). Then the function 𝑓(𝑥, ⋅) : ℝ𝑞 → ℝ is integrable on ℝ𝑞 for almost all 𝑥 ∈ ℝ𝑝 , and the function 𝑓(⋅, 𝑦) : ℝ𝑝 → ℝ is integrable on ℝ𝑝 for almost all 𝑦 ∈ ℝ𝑞 . Moreover, the equality } } { { ∫ { ∫ 𝑓(𝑥, 𝑦) 𝑑𝑦} 𝑑𝑥 = ∫ 𝑓(𝑥, 𝑦) 𝑑(𝑥, 𝑦) = ∫ { ∫ 𝑓(𝑥, 𝑦) 𝑑𝑥} 𝑑𝑦 ℝ𝑛 ℝ𝑞 {ℝ𝑝 ℝ𝑝 {ℝ𝑞 } } holds. A certain converse of the Fubini theorem is given by the following:

(0.6)

0.1 The Lebesgue integral |

9

Theorem 0.8 (Tonelli). Suppose that 𝑓 : ℝ𝑝 × ℝ𝑞 → ℝ is measurable, where 𝑝 + 𝑞 = 𝑛 and 𝑓 is written in the form 𝑧 = 𝑓(𝑥, 𝑦) (𝑥 ∈ ℝ𝑝 , 𝑦 ∈ ℝ𝑞 ). Suppose that the function 𝑓(𝑥, ⋅) : ℝ𝑞 → ℝ is integrable on ℝ𝑞 for almost all 𝑥 ∈ ℝ𝑝 , and the function 𝐹 : ℝ𝑝 → ℝ defined by 𝐹(𝑥) := ∫ |𝑓(𝑥, 𝑦)| 𝑑𝑦

(0.7)

ℝ𝑞

is integrable on ℝ𝑝 . Then 𝑓 is integrable on ℝ𝑛 . We remark that one can construct examples which show that one cannot drop the absolute value of 𝑓(𝑥, 𝑦) in (0.7). A property of functions which is particularly important (and is here supposed to hold a.e.) is boundedness. The essential supremum and essential infimum of a measur­ able function 𝑓 : 𝑀 → ℝ on 𝑀 are defined by esssup {𝑓(𝑥) : 𝑥 ∈ 𝑀} := inf sup {𝑓(𝑥) : 𝑥 ∈ 𝑀 \ 𝑁}

(0.8)

essinf {𝑓(𝑥) : 𝑥 ∈ 𝑀} := sup inf {𝑓(𝑥) : 𝑥 ∈ 𝑀 \ 𝑁} ,

(0.9)

𝜆(𝑁)=0

and 𝜆(𝑁)=0

respectively, where the infimum in (0.8) and the supremum in (0.9) are taken over all nullsets 𝑁 ⊆ 𝑀. Functions 𝑓 with finite essential supremum and essential infimum are called essentially bounded.¹ Definition 0.9. For 1 ≤ 𝑝 < ∞, the Lebesgue space L𝑝 ([𝑎, 𝑏]) consists of all measurable functions 𝑓 : [𝑎, 𝑏] → ℝ, for which 𝑏

∫ |𝑓(𝑥)|𝑝 𝑑𝑥 < ∞ .

(0.10)

𝑎

For 𝑝 = ∞, the Lebesgue space L∞ ([𝑎, 𝑏]) consists of all measurable functions 𝑓 : [𝑎, 𝑏] → ℝ which are essentially bounded on [𝑎, 𝑏]. ◼ We will consider the space L𝑝 ([𝑎, 𝑏]) with the usual norm 𝑏

‖𝑓‖L𝑝

1/𝑝

{{∫ |𝑓(𝑥)|𝑝 𝑑𝑥} for 1 ≤ 𝑝 < ∞ , := { 𝑎 {esssup {|𝑓(𝑥)| : 𝑎 ≤ 𝑥 ≤ 𝑏} for 𝑝 = ∞ .

(0.11)

To be precise, (0.11) is not a norm on L𝑝 ([𝑎, 𝑏]) since ‖𝑓‖L𝑝 = 0 does not imply that 𝑓(𝑥) ≡ 0 everywhere, but only almost everywhere on [𝑎, 𝑏]. This flaw may be overcome in two different, but equivalent ways. Either we identify any two functions 𝑓 and 𝑔 if

1 The word “essential” suggests that nullsets are negligible, i.e. 𝑓 is bounded outside some negligible set.

10 | 0 Prerequisites the set of all 𝑥 ∈ [𝑎, 𝑏] such that 𝑓(𝑥) ≠ 𝑔(𝑥) is a nullset, i.e. 𝑓 and 𝑔 coincide almost everywhere. Or (what is essentially the same), we introduce an equivalence relation on L𝑝 ([𝑎, 𝑏]) by calling two functions 𝑓 and 𝑔 equivalent (and writing 𝑓 ∼ 𝑔) if 𝑓 − 𝑔 ∈ 𝑁𝑝 ([𝑎, 𝑏]) := {ℎ ∈ L𝑝 ([𝑎, 𝑏]) : ℎ(𝑥) ≡ 0 a.e. on [𝑎, 𝑏]} .

(0.12)

In this setting, (0.11) then defines a norm on the quotient space L 𝑝 ([𝑎, 𝑏])/𝑁𝑝 ([𝑎, 𝑏]) for which we still write L𝑝 ([𝑎, 𝑏]) to avoid clumsy notation.² It is not hard to prove that (L𝑝 ([𝑎, 𝑏]), ‖ ⋅ ‖L𝑝 ) is a Banach space for each 𝑝. Throughout this book, we denote by 𝑝󸀠 the conjugate index to 𝑝 defined by³ ∞ { { { 𝑝 𝑝 := { 𝑝−1 { { {1 󸀠

if 𝑝 = 1 , (0.13)

if 1 < 𝑝 < ∞ , if 𝑝 = ∞ .

In the following proposition, we recall two important properties of the Lebesgue spaces L𝑝 ([𝑎, 𝑏]). The first property is called Hölder inequality, and the second prop­ erty shows that L𝑝 ([𝑎, 𝑏]) is strictly decreasing with respect to the index 𝑝. A certain converse of the Hölder inequality is given in Exercise 0.39. Proposition 0.10. Let 1 ≤ 𝑝 ≤ ∞ and let 𝑝󸀠 defined by (0.13). Then the following holds. (a) From 𝑓 ∈ L𝑝 ([𝑎, 𝑏]) and 𝑔 ∈ L𝑝󸀠 ([𝑎, 𝑏]), it follows that 𝑓𝑔 ∈ L1 ([𝑎, 𝑏]) with (0.14)

‖𝑓𝑔‖L1 ≤ ‖𝑓‖L𝑝 ‖𝑔‖L𝑝󸀠 . (b) The strict inclusions L∞ ([𝑎, 𝑏]) ⊂ L𝑞 ([𝑎, 𝑏]) ⊂ L𝑝 ([𝑎, 𝑏]) ⊂ L1 ([𝑎, 𝑏])

(1 < 𝑝 < 𝑞 < ∞)

(0.15)

are true. Proof. To prove (a), suppose first that 𝑝 = ∞, and hence 𝑝󸀠 = 1. Then |𝑓(𝑥)𝑔(𝑥)| ≤ ‖𝑓‖L∞ |𝑔(𝑥)| (a.e. on [𝑎, 𝑏]) implies 𝑏

𝑏

‖𝑓𝑔‖L1 = ∫ |𝑓(𝑥)𝑔(𝑥)| 𝑑𝑥 ≤ ‖𝑓‖L∞ ∫ |𝑔(𝑥)| 𝑑𝑥 = ‖𝑓‖L∞ ‖𝑔‖L1 𝑎

𝑎

which is (0.14). The case 𝑝 = 1 and 𝑝󸀠 = ∞ follows by symmetry. It remains to prove (a) for 1 < 𝑝 < ∞.

2 Therefore, to be rigorous, the linear space L𝑝 ([𝑎, 𝑏]) consists of equivalence classes of measurable functions, rather than individual functions. When working in these spaces, however, one usually treats their elements like individual functions, keeping in mind that the equality of 𝑓 and 𝑔 is meant in the sense of (0.12). For example, the characteristic function of the rational numbers in [𝑎, 𝑏] then belongs to the set 𝑁𝑝 ([𝑎, 𝑏]) for every 𝑝, and so is “equal” to the zero function. 3 Note that 𝑝󸀠󸀠 = 𝑝 for all 𝑝, and 𝑝󸀠 = 𝑝 if and only if 𝑝 = 2.

0.1 The Lebesgue integral

|

11

Consider the graph of the function 𝑥 󳨃→ 𝑥𝑝−1 for 𝑥 ≥ 0 in the (𝑥, 𝑦)-plane which 󸀠 coincides with the graph of 𝑦 󳨃→ 𝑦𝑝 −1 for 𝑦 ≥ 0 in the (𝑦, 𝑥)-plane. Fix 𝜉, 𝜂 > 0, denote by 𝐴 the area of the region between this graph, the 𝑥-axis, and the line 𝑥 = 𝜉, and by 𝐵, the area of the region between this graph, the 𝑦-axis, and the line 𝑦 = 𝜂. A simple geometric reasoning then shows that 𝜂

𝜉 𝑝−1

𝜉𝜂 ≤ 𝐴 + 𝐵 = ∫ 𝑥

󸀠

󸀠

𝑑𝑥 + ∫ 𝑦𝑝 −1 𝑑𝑦 =

0

0

𝜉𝑝 𝜂𝑝 + 󸀠 . 𝑝 𝑝

(0.16)

Now, given 𝑓 ∈ L𝑝 ([𝑎, 𝑏]) and 𝑔 ∈ L𝑝󸀠 ([𝑎, 𝑏]), we put⁴ ̂ := 𝑓(𝑥) , 𝑓(𝑥) ‖𝑓‖L𝑝

̂ 𝑔(𝑥) :=

𝑔(𝑥) . ‖𝑔‖L𝑝󸀠

̂ ̂ and 𝜂 := |𝑔(𝑥)| in (0.16) yields Then ‖𝑓‖̂ L𝑝 = ‖𝑔‖̂ L𝑝󸀠 = 1, and taking 𝜉 := |𝑓(𝑥)| 𝑏

𝑏

𝑏

𝑎

𝑎

𝑎

󸀠

̂ 𝑝 ̂ 𝑝 |𝑓(𝑥)| |𝑔(𝑥)| 1 1 ̂ 𝑔(𝑥)| ̂ ̂ = ∫ |𝑓(𝑥) ̂ 𝑑𝑥 + 𝑑𝑥 ≤ ∫ ∫ 𝑑𝑥 = + 󸀠 = 1 . ‖𝑓𝑔‖ L1 󸀠 𝑝 𝑝 𝑝 𝑝 Taking into account the definition of 𝑓 ̂ and 𝑔̂ in this estimate and multiplying by ‖𝑓‖L𝑝 ‖𝑔‖L 󸀠 gives (0.14). 𝑝 To prove (b), we apply (a) to the special choice 𝑔(𝑥) ≡ 1. For 𝑓 ∈ L∞ ([𝑎, 𝑏]), we then get 1/𝑞

𝑏 𝑞

‖𝑓‖L𝑞 = (∫ |𝑓(𝑥)| 𝑑𝑥) 𝑎

≤ (𝑏 − 𝑎)1/𝑞 esssup {|𝑓(𝑥)| : 𝑎 ≤ 𝑥 ≤ 𝑏} = (𝑏 − 𝑎)1/𝑞 ‖𝑓‖L∞ . Similarly, for 𝑝 < 𝑞 and 𝑓 ∈ L𝑞 ([𝑎, 𝑏]), we get 1/𝑝

𝑏 𝑝

‖𝑓‖L𝑝 = (∫ |𝑓(𝑥)| 𝑑𝑥) 𝑎 1/𝑞

𝑏 (𝑞−𝑝)/𝑝𝑞

≤ (𝑏 − 𝑎)

𝑞

(∫ |𝑓(𝑥)| 𝑑𝑥)

= (𝑏 − 𝑎)(𝑞−𝑝)/𝑝𝑞 ‖𝑓‖L𝑞 ,

𝑎

4 To be precise, we have to ensure that neither 𝑓 nor 𝑔 is zero a.e.; however, in this case, both the right-hand and left-hand side of (0.14) vanish, and there is nothing to prove.

12 | 0 Prerequisites while for 𝑝 > 1 and 𝑓 ∈ L𝑝 ([𝑎, 𝑏]), we get 𝑏

‖𝑓‖L1 = ∫ |𝑓(𝑥)| 𝑑𝑥 𝑎 1/𝑝

𝑏

≤ (𝑏 − 𝑎)(𝑝−1)/𝑝 (∫ |𝑓(𝑥)|𝑝 𝑑𝑥)

= (𝑏 − 𝑎)(𝑝−1)/𝑝 ‖𝑓‖L𝑝

𝑎

which completes the proof. We still have to prove that the inclusions in (0.15) are strict. For finite 𝑞, the function {0 𝑓(𝑥) := { 1 { (𝑥−𝑎)𝛼

for 𝑥 = 𝑎 , for 𝑎 < 𝑥 ≤ 𝑏 ,

where 𝛼 ∈ (0, 1/𝑞), may serve as an example of a function 𝑓 ∈ L𝑞 ([𝑎, 𝑏]) \ L∞ ([𝑎, 𝑏]). For the other inclusions, we now construct some more sophisticated examples which build on the following well-known fact from a first year calculus course: for 𝛼, 𝛽 ∈ ℝ, the series ∞ 1 𝜁(𝛼, 𝛽) := ∑ (0.17) 𝛼 log𝛽 (𝑛 + 1) 𝑛 𝑛=1 converges if and only if either 𝛼 > 1 and 𝛽 is arbitrary (in particular, 𝛽 = 0), or 𝛼 = 1 and 𝛽 > 1. This may be easily proved by means of the classical condensation theorem for series with decreasing terms. Example 0.11. For 1 ≤ 𝑝 < ∞, we construct a function 𝑓 ∈ L𝑝 ([0, 1]) \ (⋃ L𝑞 ([0, 1])) . 𝑞>𝑝

Let 𝑓(0) := 0 and 𝑓(𝑥) :=

𝑛1/𝑝 log2 (𝑛 + 1)

(

1 1 𝑝, we have 1

∞

∫ |𝑓(𝑥)|𝑞 𝑑𝑥 = ∑ 0

𝑛=1

1/𝑛

∫ 1/(𝑛+1)

∞ 𝑛𝑞/𝑝 𝑛(𝑞−𝑝)/𝑝 𝑑𝑥 = ∑ = ∞, 2𝑞 log (𝑛 + 1) 𝑛=1 (𝑛 + 1) log (𝑛 + 1) 2𝑞

which shows that 𝑓 ∈ ̸ L𝑞 ([0, 1]) for any 𝑞 > 𝑝.

♥

0.1 The Lebesgue integral

|

13

Example 0.12. For 1 < 𝑝 ≤ ∞, we construct a function 𝑓 ∈ (⋂ L𝑞 ([0, 1])) \ L𝑝 ([0, 1]) . 𝑞 1. Similarly, by Example 0.13, we may find a function ℎ ∈ L𝑝 ([1, ∞)) \ (⋃ L𝑠 ([1, ∞))) 𝑠 0 for 𝑡 > 0, and 𝜙(𝑡) → ∞ as 𝑡 → ∞. Given a Young function 𝜙 : [0, ∞) → [0, ∞), we write 𝑓 ∈ L𝜙 ([𝑎, 𝑏]) if 𝑓 : [𝑎, 𝑏] → ℝ is measurable and 𝑏

∫ 𝜙(|𝑓(𝑥)|) 𝑑𝑥 < ∞ .

(0.18)

𝑎

The set L𝜙 ([𝑎, 𝑏]) is called the Orlicz class generated by 𝜙.

◼

Typical examples of Young functions are 𝜙(𝑡) = 𝑡𝑝 for 1 ≤ 𝑝 < ∞, 𝜙(𝑡) = 𝑒𝑡 − 1, or 𝜙(𝑡) = (𝑡 + 1) log(𝑡 + 1). For the choice 𝜙(𝑡) = 𝑡𝑝 with 1 ≤ 𝑝 < ∞, the Orlicz class L𝜙 ([𝑎, 𝑏]) coincides, of course, with the Lebesgue space L𝑝 ([𝑎, 𝑏]), as a comparison of (0.10) and (0.18) shows. Unfortunately, for general Young functions 𝜙, the Orlicz class need not be a linear space, as the following simple example shows. Example 0.17. Let 𝜙(𝑡) = 𝑒𝑡 − 1, and let 𝑓 : [0, 1] → ℝ be defined by {− 1 log 𝑥 𝑓(𝑥) := { 2 0 {

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 .

6 We define a Young function 𝜙 here on the half-axis [0, ∞); therefore, we have to write 𝜙(|𝑓(𝑥)|) if 𝑓 is a real function as in (0.18). To avoid the absolute value in the argument, some authors consider Young functions 𝜙 on the whole real axis by defining them first on [0, ∞) and extending them afterwards as even functions to (−∞, 0]. As we will show in Lemma 1.36 in the next chapter, a Young function is always increasing.

16 | 0 Prerequisites Then

1

1

∫ 𝜙(|𝑓(𝑥)|) 𝑑𝑥 = ∫ [ 0

0

1 − 1] 𝑑𝑥 < ∞ , √𝑥

so 𝑓 ∈ L𝜙 ([𝑎, 𝑏]), but 1

1

∫ 𝜙(2|𝑓(𝑥)|) 𝑑𝑥 = ∫ [ 0

0

1 − 1] 𝑑𝑥 = ∞ , 𝑥

so 2𝑓 ∈ ̸ L𝜙 ([𝑎, 𝑏]).

♥

The reason for the phenomenon described in Example 0.17 will become clear later (Proposition 0.20). One may associate to every Orlicz class L𝜙 ([𝑎, 𝑏]), a linear space, called Orlicz space, in the following way. First, we observe that for any Young function 𝜙, the set 𝑏

𝐴(𝜙) := {𝑓 ∈ L𝜙 ([𝑎, 𝑏]) : ∫ 𝜙(|𝑓(𝑥)|) 𝑑𝑥 ≤ 1}

(0.19)

𝑎

is convex, symmetric, balanced, and absorbing.⁷ Therefore, the Minkowski functional associated to 𝐴(𝜙), i.e. 𝜇𝐴(𝜙) (𝑓) := inf {𝜆 > 0 : 𝑓/𝜆 ∈ 𝐴(𝜙)}

(0.20)

is a norm on the linear space of all measurable functions 𝑓 : [𝑎, 𝑏] → ℝ with the property that 𝑓/𝜆 ∈ L𝜙 ([𝑎, 𝑏]) for some 𝜆 > 0. Moreover, the closed unit ball in this norm coincides with the set 𝐴(𝜙) given in (0.19). Definition 0.18. We call this linear space the Orlicz space L𝜙 ([𝑎, 𝑏]) generated by the Young function 𝜙. In the sequel, we write ‖𝑓‖L𝜙 for (0.20) and call ‖𝑓‖L𝜙 the Luxemburg norm of 𝑓. ◼ The natural question arises for what Young functions we have L𝜙 ([𝑎, 𝑏]) = L𝜙 ([𝑎, 𝑏]), which means that the Orlicz class L𝜙 ([𝑎, 𝑏]) itself is a linear space. This question may be answered by means of the following notion. Definition 0.19. A Young function 𝜙 : [0, ∞) → [0, ∞) satisfies a 𝛥 2 -condition if 𝜙(2𝑡) ≤ 𝑀𝜙(𝑡)

(𝑡 ≥ 𝑇)

for suitable constants 𝑀 > 0 and 𝑇 > 0. In this case, we write 𝜙 ∈ 𝛥 2 .

(0.21) ◼

Proposition 0.20. The Orlicz class L𝜙 ([𝑎, 𝑏]) is a linear space, and hence coincides with L𝜙 ([𝑎, 𝑏]), if and only if 𝜙 ∈ 𝛥 2 .

7 The definition of these notions may be found in many textbooks on Functional Analysis, e.g. [270].

0.1 The Lebesgue integral

|

17

We do not prove Proposition 0.20; the proof may be found, for example, in the book [169]. Instead, let us look at some examples. The Young function 𝜙(𝑡) = 𝑡𝑝 (1 ≤ 𝑝 < ∞) obviously satisfies the 𝛥 2 -condition (0.21) for arbitrary 𝑇 > 0 and 𝑀 := 2𝑝 , and so L𝜙 ([𝑎, 𝑏]) = L𝜙 ([𝑎, 𝑏]) = L𝑝 ([𝑎, 𝑏]) in this case. Moreover, here, the Minkowski func­ tional (0.20) coincides with the L𝑝 norm (0.11) because 𝑏

} { 𝑓(𝑥) ) 𝑑𝑥 ≤ 1} 𝜇𝐴(𝜙) (𝑓) = inf {𝜆 > 0 : ∫ 𝜙 ( 𝜆 𝑎 } { 𝑏

{ } = inf {𝜆 > 0 : ∫ |𝑓(𝑥)|𝑝 𝑑𝑥 ≤ 𝜆𝑝 } = ‖𝑓‖L𝑝 . 𝑎 { } On the other hand, the Young function 𝜙(𝑡) = 𝑒𝑡 − 1 from Example 0.17 cannot satisfy a 𝛥 2 -condition since the corresponding Orlicz class L𝜙 ([𝑎, 𝑏]) is not a linear space, as we have seen there. In fact, lim

𝑡→∞

𝜙(2𝑡) 𝑒2𝑡 − 1 𝑒𝑡 − 𝑒−𝑡 = lim 𝑡 = lim = ∞, 𝑡→∞ 𝑒 − 1 𝑡→∞ 1 − 𝑒−𝑡 𝜙(𝑡)

(0.22)

and so 𝜙 ∈ ̸ 𝛥 2 . It is illuminating to calculate the norm ‖𝑓‖L𝜙 for 𝑓 and 𝜙 from Example 0.17. For any 𝜆 > 0, we have 1

1

∫ 𝜙(|𝑓(𝑥)|/𝜆) 𝑑𝑥 = ∫ [ 0

0

1 − 1] 𝑑𝑥 < ∞ 𝑥1/𝜆

if and only if 𝜆 > 1, and in this case, for the integral, we get 1

∫[ 0

1 𝑥1/𝜆

− 1] 𝑑𝑥 = [

1 𝜆 1−1/𝜆 1 𝑥 . − 𝑥] = 𝜆−1 𝜆 − 1 0

The smallest value of 𝜆 for which the last expression is ≤ 1 is ‖𝑓‖L𝜙 = 𝜆 = 2. We have defined L𝜙 ([𝑎, 𝑏]) as the set (actually, linear space) of all measurable func­ tions 𝑓 : [𝑎, 𝑏] → ℝ such that 𝑓/𝜆 ∈ L𝜙 ([𝑎, 𝑏]) for some 𝜆 > 0. It is also interesting to consider the set 𝐸𝜙 ([𝑎, 𝑏]) of all measurable functions 𝑓 : [𝑎, 𝑏] → ℝ such that 𝑓/𝜆 ∈ L𝜙 ([𝑎, 𝑏]) for all 𝜆 > 0; this is usually called the small Orlicz space. Equipped with the norm (0.20), this is a closed linear subspace of L𝜙 ([𝑎, 𝑏]) which coincides with L𝜙 ([𝑎, 𝑏]) if and only if 𝜙 ∈ 𝛥 2 . This again illustrates the importance of the 𝛥 2 -condi­ tion (0.21). Further important properties of the space 𝐸𝜙 will be considered in Exer­ cises 0.22–0.24. In rather the same way as we defined in (0.13) the conjugate index of a number 𝑝, we may associate to each Young function some kind of conjugate Young function. Given a Young function 𝜙 : [0, ∞) → [0, ∞), we define its conjugate Young function 𝜙∗ : [0, ∞) → [0, ∞) by 𝜙∗ (𝑡) := sup {𝑠𝑡 − 𝜙(𝑠) : 𝑠 ≥ 0} . (0.23)

18 | 0 Prerequisites By construction, we then have 𝜙∗∗ = 𝜙 and 𝑎𝑏 ≤ 𝜙(𝑎) + 𝜙∗ (𝑏)

(𝑎, 𝑏 ≥ 0) ;

(0.24)

inequality (0.24) is usually referred to as Young’s inequality. For example, an easy cal­ culation shows that the conjugate Young function to 𝜙(𝑡) :=

1 𝑝 𝑡 𝑝

for 1 < 𝑝 < ∞ is 𝜙∗ (𝑡) :=

1 𝑝󸀠 𝑡 , 𝑝󸀠

where 𝑝󸀠 is given by (0.13). In this sense, the conjugate Young function plays the same role for Orlicz spaces as the conjugate index for Lebesgue spaces. Young’s inequality (0.24) may then be considered as an analogue to Hölder’s inequality (0.14).

0.2 Some functional analysis In the following chapters, we will need three concepts from the theory of Banach spaces: equivalence of norms, imbeddings between spaces, and duality. We start with the concept of duality which is of fundamental importance in functional analysis and operator theory. Recall that the dual space 𝑋∗ of a real normed linear space consists of all bounded linear functionals⁸ ℓ : 𝑋 → ℝ, equipped with the norm⁹ ‖ℓ‖𝑋∗ := sup {|⟨𝑥, ℓ⟩| : ‖𝑥‖𝑋 ≤ 1} = sup {|⟨𝑥, ℓ⟩| : ‖𝑥‖𝑋 = 1} .

(0.25)

One of the most important results on dual spaces is the Hahn–Banach theorem which asserts that a bounded linear functional on a subspace may be extended to the whole space without “increasing its size.” For measuring the size of a functional, we recall that a sublinear functional is a map 𝑝 : 𝑋 → ℝ with the property that 𝑝(𝑥 + 𝑦) ≤ 𝑝(𝑥) + 𝑝(𝑦),

𝑝(𝜆𝑥) = 𝜆𝑝(𝑥) (𝑥, 𝑦 ∈ 𝑋; 0 ≤ 𝜆 < ∞) .

(0.26)

A standard example is 𝑝(𝑥) = ‖𝑥‖, but more sophisticated choices of 𝑝 lead to in­ teresting results. With this notion, the announced extension theorem reads as follows.

8 Since we only use real vector spaces in this book, all functionals are considered real-valued. For this reason, we only recall the real version of the Hahn–Banach theorem. 9 Here, we adopt the usual notation ⟨𝑥, ℓ⟩ instead of ℓ(𝑥) to emphasize the idea of duality pairing borrowed from scalar products in Hilbert space. We remark that the supremum in (0.25) could also be taken over the open unit ball, see Exercise 0.40.

0.2 Some functional analysis

| 19

Theorem 0.21 (Hahn–Banach). Let 𝑋 be a real normed space and 𝑝 be some sublinear functional on 𝑋. Let 𝑈 ⊂ 𝑋 be a linear subspace, and suppose that a functional ℓ ∈ 𝑈∗ satisfies ⟨𝑢, ℓ⟩ ≤ 𝑝(𝑢) for all 𝑢 ∈ 𝑈. Then there exists a functional ℓ̂ ∈ 𝑋∗ such that ℓ|̂ 𝑈 = ℓ, i.e. ⟨𝑢, ℓ⟩̂ = ⟨𝑢, ℓ⟩ (𝑢 ∈ 𝑈) and ⟨𝑥, ℓ⟩̂ ≤ 𝑝(𝑥) for all 𝑥 ∈ 𝑋. We do not prove Theorem 0.21 since it is beyond the scope of this book. Instead, we prove three interesting consequences of this theorem which we will need in Chapter 4. Corollary 0.22. Let 𝑋 be a real normed space and 𝑈 ⊂ 𝑋 be a linear subspace. Then the following is true. (a) Every functional ℓ ∈ 𝑈∗ may be extended to a functional ℓ̂ ∈ 𝑋∗ such that ℓ|̂ 𝑈 = ℓ and ‖ℓ‖̂ 𝑋∗ = ‖ℓ‖𝑈∗ . (b) Given any 𝑥∗ ∈ 𝑋 \ {0}, one may find a functional ℓ̂ ∈ 𝑋∗ such that ‖ℓ‖̂ 𝑋∗ = 1 and ⟨𝑥∗ , ℓ⟩̂ = ‖𝑥∗ ‖𝑋 . (c) The equality ‖𝑥‖𝑋 = sup {|⟨𝑥, ℓ⟩| : ‖ℓ‖𝑋∗ ≤ 1} = sup {|⟨𝑥, ℓ⟩| : ‖ℓ‖𝑋∗ = 1} ,

(0.27)

which is dual to (0.25), holds true. Proof. (a) It follows from the properties of a norm¹⁰ that the map 𝑝(𝑥) := ‖ℓ‖𝑈∗ ‖𝑥‖𝑋 is a sublinear functional on 𝑋. Moreover, (0.25) implies that ⟨𝑢, ℓ⟩ ≤ 𝑝(𝑢) for all 𝑢 ∈ 𝑈. So, from Theorem 0.21, we conclude that ℓ admits an extension ℓ̂ ∈ 𝑋∗ satisfying ⟨𝑥, ℓ⟩̂ ≤ 𝑝(𝑥) = ‖ℓ‖𝑈∗ ‖𝑥‖𝑋

(𝑥 ∈ 𝑋) ,

and hence ‖ℓ‖̂ 𝑋∗ ≤ ‖ℓ‖𝑈∗ , again by (0.25). The reverse estimate ‖ℓ‖̂ 𝑋∗ ≥ ‖ℓ‖𝑈∗ is obvi­ ous. (b) On the one-dimensional subspace 𝑈 := ℝ𝑥∗ = {𝜆𝑥∗ : 𝜆 ∈ ℝ} ⊂ 𝑋, define ℓ ∈ 𝑈∗ by ⟨𝜆𝑥∗ , ℓ⟩ := 𝜆‖𝑥∗ ‖𝑋 . Clearly, ⟨𝑥∗ , ℓ⟩ = ‖𝑥∗ ‖𝑋 . We claim that ‖ℓ‖𝑈∗ = 1. In fact, this follows from the equalities ‖ℓ‖𝑈∗ = sup {|⟨𝜆𝑥∗ , ℓ⟩| : ‖𝜆𝑥∗ ‖ ≤ 1} = sup 𝜆>0

𝜆‖𝑥∗ ‖𝑋 = 1. ‖𝜆𝑥∗ ‖𝑋

∗

By (a), we may now extend ℓ to a functional ℓ̂ ∈ 𝑋 which still has norm ‖ℓ‖̂ 𝑋∗ = 1, and (b) is proved. (c) From the obvious estimate |⟨𝑥, ℓ⟩| ≤ ‖ℓ‖𝑋∗ ‖𝑥‖𝑋 which holds for all 𝑥 ∈ 𝑋 and ℓ ∈ 𝑋∗ , it follows that ‖𝑥‖𝑋 ≥ sup {|⟨𝑥, ℓ⟩| : ‖ℓ‖𝑋∗ = 1} .

(0.28)

10 In this proof, we equip every norm with a subscript since we have to consider norms in the four different spaces 𝑋, 𝑈, 𝑋∗ , and 𝑈∗ . Later, we will drop the subscript when the underlying space is clear.

20 | 0 Prerequisites To prove the converse estimate, fix 𝑥∗ ∈ 𝑋 (without loss of generality, ‖𝑥∗ ‖𝑋 = 1). By statement (b), we may then find a functional ℓ ∈ 𝑋∗ such that ‖ℓ‖𝑋∗ = 1 and ⟨𝑥∗ , ℓ⟩ = ‖𝑥∗ ‖𝑋 . In other words, for this functional ℓ, the supremum in (0.28) is at­ tained, and so we get equality in (0.28). An important problem in functional analysis is to identify the dual space 𝑋∗ of a given space 𝑋. In some cases, this is very easy. For example, it is not hard to see that the dual space of 𝑋 = ℝ𝑛 is again 𝑋∗ = ℝ𝑛 . More precisely, the map 𝛷 : ℝ𝑛 → (ℝ𝑛 )∗ defined by 𝛷(𝑦) := ℓ𝑦 with 𝑛

⟨𝑥, ℓ𝑦 ⟩ := ∑ 𝑥𝑘 𝑦𝑘

(𝑥 ∈ ℝ𝑛 )

𝑘=1

is a linear surjective isometry¹¹ where the term “isometry” means that ‖ℓ𝑦 ‖𝑋∗ = ‖𝑦‖𝑋 with ‖ℓ𝑦 ‖𝑋∗ denoting the functional norm (0.25) on ℝ𝑛 . A more interesting (and very important) example is the Lebesgue space 𝑋 = L𝑝 ([𝑎, 𝑏]) which we considered in the first section. Without going into details, we state the corresponding result in the following Theorem 0.23. For 1 ≤ 𝑝 < ∞, let 𝑝󸀠 denote the conjugate index (0.13) to 𝑝 (in particular, 𝑝󸀠 = ∞ for 𝑝 = 1). For fixed 𝑔 ∈ L𝑝󸀠 ([𝑎, 𝑏]), we define a functional ℓ𝑔 : L𝑝 ([𝑎, 𝑏]) → ℝ by 𝑏

⟨𝑓, ℓ𝑔 ⟩ := ∫ 𝑓(𝑡)𝑔(𝑡) 𝑑𝑡

(𝑓 ∈ L𝑝 ([𝑎, 𝑏])) .

(0.29)

𝑎

From Hölder’s inequality (0.14), it follows that ℓ𝑔 ∈ L∗𝑝 . The interesting point is that all elements ℓ ∈ L∗𝑝 have this form: Theorem 0.23 (Riesz). For 1 ≤ 𝑝 < ∞, the dual space L∗𝑝 of the space L𝑝 with norm (0.11) may be identified with the space L𝑝󸀠 . More precisely, the map 𝛷 : L𝑝󸀠 ([𝑎, 𝑏]) → L𝑝 ([𝑎, 𝑏])∗ defined by 𝛷(𝑔) := ℓ𝑔 , with ℓ𝑔 as in (0.29) for 𝑓 ∈ L𝑝 ([𝑎, 𝑏]) and 𝑔 ∈ L𝑝󸀠 ([𝑎, 𝑏]), is a linear surjective isometry. As before, the term “isometry” means that ‖ℓ𝑔 ‖L∗𝑝 = ‖𝑔‖L𝑝󸀠 , where the norm on the dual space is given by (0.25). In an explicit form, this means that 1/𝑝󸀠

𝑏 𝑝

󸀠

(∫ |𝑔(𝑥)| 𝑑𝑥) 𝑎

𝑏

𝑏

{ } = sup {∫ 𝑓(𝑥)𝑔(𝑥) 𝑑𝑥 : ∫ |𝑓(𝑥)|𝑝 𝑑𝑥 ≤ 1} 𝑎 {𝑎 }

(0.30)

for 1 < 𝑝 < ∞, and 𝑏

𝑏

} { esssup {|𝑔(𝑥)| : 𝑎 ≤ 𝑥 ≤ 𝑏} = sup {∫ 𝑓(𝑥)𝑔(𝑥) 𝑑𝑥 : ∫ |𝑓(𝑥)| 𝑑𝑥 ≤ 1} 𝑎 } {𝑎

(0.31)

11 Here, it is important to consider both spaces 𝑋 and 𝑋∗ equipped with the Euclidean norm; if we equip 𝑋 with another norm, then we have to choose another norm on 𝑋∗ to make 𝛷 an isometry, see Exercises 0.42 and 0.43.

0.2 Some functional analysis

| 21

in case 𝑝 = 1. Interestingly, the map 𝛷 is also an isometry between L1 and L∗∞ , which means that 𝑏

𝑏

{ } (0.32) ∫ |𝑔(𝑥)| 𝑑𝑥 = sup {∫ 𝑓(𝑥)𝑔(𝑥) 𝑑𝑥 : esssup {|𝑓(𝑥)| : 𝑎 ≤ 𝑥 ≤ 𝑏} ≤ 1} . 𝑎 {𝑎 } However, Theorem 0.23 is not true in case 𝑝 = ∞ which means that L∗∞ ≠ L1 . The reason is that there are bounded linear functionals on L∞ which cannot be written in the form ℓ𝑔 as in (0.29), and so 𝛷 : L1 ([𝑎, 𝑏]) → L∗∞ ([𝑎, 𝑏]) is not surjective. It is also interesting to observe that the formulas (0.30)–(0.32) may be made more precise by constructing functions for which the supremum in these formulas is actu­ ally achieved (Exercises 0.31, 0.33, and 0.36). Theorem 0.23 shows that L∗𝑝 = L𝑝/(𝑝−1) for 1 ≤ 𝑝 < ∞, where the duality between ∗ L𝑝 and L𝑝/(𝑝−1) is given by (0.29). One could ask whether or not (0.29) also gives the gen­ eral form of a bounded linear functional on the space 𝐶([𝑎, 𝑏]), where now, of course, both functions 𝑓 and 𝑔 have to be chosen continuous. The following example shows that the answer is negative: Example 0.24. Let ℓ : 𝐶([0, 1]) → ℝ be the evaluation functional at 0, i.e. the map defined by ⟨𝑓, ℓ⟩ := 𝑓(0). Clearly, ℓ is both linear and continuous, so ℓ ∈ 𝐶∗ with ‖ℓ‖𝐶∗ = 1. However, there is no 𝑔 ∈ 𝐶([0, 1]) such that ℓ may be represented in the form (0.29). To see this, suppose that there exists a continuous function 𝑔 : [0, 1] → ℝ satisfy­ ing 1

∫ 𝑓(𝑡)𝑔(𝑡) 𝑑𝑡 = 𝑓(0)

(𝑓 ∈ 𝐶([0, 1])) .

(0.33)

0

Consider the sequence (𝑓𝑛 )𝑛 of continuous peak functions defined by {1 − 𝑛𝑥 for 0 ≤ 𝑥 < 𝑛1 , 𝑓𝑛 (𝑥) := max {1 − 𝑛𝑥, 0} = { 0 for 𝑛1 ≤ 𝑥 ≤ 1 . { Then 𝑓𝑛 (0) = 1 for all 𝑛, and 𝑓𝑛 → 𝑓 := 𝜒{0} pointwise on [0, 1]. Moreover,¹² 1

1

∫ 𝑓𝑛 (𝑡)𝑔(𝑡) 𝑑𝑡 ≤ ‖𝑔‖𝐶 ∫ 𝑓𝑛 (𝑡) 𝑑𝑡 = 0

0

‖𝑔‖𝐶 . 2𝑛

Finally, the majorization (0.3) holds with 𝑓𝑛 replaced by 𝑓𝑛 𝑔 and the majorant function 𝐹(𝑥) ≡ ‖𝑔‖𝐶 on 𝑀 = [0, 1]. Thus, Theorem 0.4 implies that 1

1

1 = 𝑓(0) = ∫ 𝑓(𝑡)𝑔(𝑡) 𝑑𝑡 = lim ∫ 𝑓𝑛 (𝑡)𝑔(𝑡) 𝑑𝑡 = 0 , 𝑛→∞

0

0

a palpable contradiction. 12 The norm ‖ ⋅ ‖𝐶 in the space of continuous functions is defined in (0.45) below.

♥

22 | 0 Prerequisites In Chapter 4, we will give a precise description of the dual space 𝐶∗ of 𝐶, and this is one of the places where functions of bounded variation play a very prominent role, see Theorem 4.31. Apart from the description of the dual space 𝑋∗ of a given normed space 𝑋, an­ other important problem consists of finding properties which carry over from 𝑋 to 𝑋∗ or vice versa. As a sample result, we discuss separability. Definition 0.25. A normed space 𝑋 is called separable if it contains a dense countable ◼ subset, i.e. a set 𝑀 = {𝑥1 , 𝑥2 , 𝑥3 , . . .} satisfying 𝑀 = 𝑋. Here are some examples of separable and nonseparable spaces. Clearly, ℝ𝑁 is sepa­ rable with any norm (choose 𝑀 := ℚ𝑁 ). Moreover, 𝐶([𝑎, 𝑏]) is separable by the clas­ sical Weierstrass approximation theorems.¹³ It is much harder to prove that the space L𝑝 ([𝑎, 𝑏]) is also separable for 1 ≤ 𝑝 < ∞; a proof may be found, for example, in [61]. However, it is easy to see that L∞ ([𝑎, 𝑏]) is not separable: Example 0.26. For any 𝑐 ∈ (𝑎, 𝑏), let 𝑓𝑐 := 𝜒[𝑎,𝑐] denote the characteristic function of the interval [𝑎, 𝑐]. Clearly, ‖𝑓𝑐 − 𝑓𝑑 ‖L∞ = 1 for 𝑎 < 𝑐 < 𝑑 < 𝑏. This shows that the open balls 𝐵𝑐 := {𝑓 ∈ L∞ ([𝑎, 𝑏]) : ‖𝑓 − 𝑓𝑐 ‖L∞ < 1/2} (𝑎 < 𝑐 < 𝑏) are mutually disjoint for different values of 𝑐, and there are uncountably many of them. However, this immediately implies that L∞ ([𝑎, 𝑏]) cannot be separable. ♥ Example 0.26 and Theorem 0.23 (for 𝑝 = 1) show that the separability of 𝑋 does not imply the separability of 𝑋∗ . The converse, however, is true: Theorem 0.27. If 𝑋∗ is separable, then 𝑋 is also separable. Proof. Let {ℓ1̃ , ℓ2̃ , ℓ3̃ , . . .} be dense in 𝑋∗ . Normalizing¹⁴ ℓ𝑛 := ℓ𝑛̃ /‖ℓ𝑛̃ ‖𝑋∗ , we then get a dense subset {ℓ1 , ℓ2 , ℓ3 , . . .} of the unit sphere in 𝑋∗ . By definition of the functional norm (0.25), we can find a sequence (𝑥𝑛 )𝑛 in the unit sphere of 𝑋 such that |ℓ𝑛 (𝑥𝑛 )| > 1/2 for all 𝑛 ∈ ℕ. Denote by 𝑀 the rational span of {𝑥1 , 𝑥2 , 𝑥3 , . . .}, i.e. the set of all linear combinations of the 𝑥𝑛 ’s with rational coefficients. Clearly, 𝑀 is countable; we claim that 𝑀 is dense in 𝑋. In fact, assuming the contrary, we can find an element ℓ ∈ 𝑋∗ satisfying ‖ℓ‖𝑋∗ = 1 and vanishing on 𝑀. In particular, we then have ℓ(𝑥𝑛 ) = 0 for all 𝑛. Choose ℓ𝑚 from the set {ℓ1 , ℓ2 , ℓ3 , . . .} defined above such that ‖ℓ𝑚 − ℓ‖𝑋∗ < 1/2, which is possible since

13 To be precise, we first approximate a fixed continuous function on [𝑎, 𝑏] uniformly by a sequence of polynomials, and then each of these polynomials by a polynomial of the same degree, but with rational coefficients. Thus, we choose as 𝑀 here the (countable) set of all polynomials with rational coefficients. 14 Obviously, it is no loss of generality to assume that all functionals ℓ𝑛̃ are different from zero.

0.2 Some functional analysis

| 23

this set is dense in the unit sphere in 𝑋∗ . Then ‖𝑥𝑚 ‖𝑋 = 1 and ℓ(𝑥𝑚 ) = 0 imply that 1 1 < |ℓ𝑚 (𝑥𝑚 )| ≤ |ℓ𝑚 (𝑥𝑚 ) − ℓ(𝑥𝑚 )| + |ℓ(𝑥𝑚 )| ≤ ‖ℓ𝑚 − ℓ‖𝑋∗ < , 2 2 a contradiction. Now, we recall the definition of equivalent norms and continuous imbeddings. We start with the following Definition 0.28. Two norms ‖ ⋅ ‖𝑋 and ⦀ ⋅ ⦀𝑋 on a linear space 𝑋 are called equivalent if there exist constants 𝑀, 𝑚 > 0 such that 𝑚‖𝑓‖𝑋 ≤ ⦀𝑓⦀𝑋 ≤ 𝑀‖𝑓‖𝑋 for all 𝑓 ∈ 𝑋.

(0.34) ◼

Geometrically, the equivalence of two norms means that every ball with respect to one of these norms may be included, possibly after diminishing its radius into a ball with respect to the other norm. We will consider various examples of equivalent and nonequivalent norms below. Since convergent sequences and Cauchy sequences are the same with respect to two equivalent norms, a Banach space remains complete when passing to an equiva­ lent norm. There is a nontrivial converse of this statement which states that whenever a linear space 𝑋 is complete with respect to two norms ‖ ⋅ ‖𝑋 and ⦀ ⋅ ⦀𝑋 , and one of the estimates in (0.34) holds, then the other estimate also holds.¹⁵ Definition 0.29. We say that a normed linear space (𝑋, ‖ ⋅ ‖𝑋 ) is imbedded¹⁶ into an­ other normed linear space (𝑌, ‖ ⋅ ‖𝑌 ) if 𝑋 ⊆ 𝑌 and ‖𝑓‖𝑌 ≤ 𝑐‖𝑓‖𝑋

(𝑓 ∈ 𝑋)

(0.35)

for some constant 𝑐 > 0 independent of 𝑓. In this case, we write 𝑋 󳨅→ 𝑌 and call 𝑐 in (0.35) an imbedding constant. Moreover, the smallest possible imbedding constant 𝑐, i.e. 𝑐(𝑋, 𝑌) := sup {‖𝑓‖𝑌 : ‖𝑓‖𝑋 ≤ 1} , (0.36) will be called the sharp imbedding constant for 𝑋 󳨅→ 𝑌 in the sequel.

◼

The importance (and usefulness) of the imbedding condition (0.35) consists of the fact that 𝑓𝑛 → 𝑓 in 𝑋 implies 𝑓𝑛 → 𝑓 in 𝑌. If merely 𝑋 ⊆ 𝑌, but with unrelated norms, this need not be true.

15 This result is a consequence of the well-known closed graph theorem (or open mapping theorem, or inverse mapping theorem) in functional analysis, applied to the identity operator between (𝑋, ‖ ⋅ ‖𝑋 ) and (𝑋, ⦀ ⋅ ⦀𝑋 ). 16 Some authors state more precisely that (𝑋, ‖⋅‖𝑋 ) is continuously imbedded into (𝑌, ‖⋅‖𝑌 ) since (0.35) means that the identity operator is continuous between these spaces.

24 | 0 Prerequisites Now, we recall some definitions and facts about Banach spaces whose elements can be multiplied. Definition 0.30. A normed linear space (𝑋, ‖ ⋅ ‖𝑋 ) is called an algebra if the product of two elements 𝑓, 𝑔 ∈ 𝑋 also belongs to 𝑋 and satisfies ‖𝑓𝑔‖𝑋 ≤ 𝑐‖𝑓‖𝑋 ‖𝑔‖𝑋

(𝑓, 𝑔 ∈ 𝑋)

(0.37)

for some constant 𝑐 > 0 independent of 𝑓 and 𝑔. If (𝑋, ‖ ⋅ ‖𝑋 ) is complete, 𝑋 is called a Banach algebra. ◼ Sometimes, (0.37) may be sharpened to ‖𝑓𝑔‖𝑋 ≤ ‖𝑓‖𝑋 ‖𝑔‖𝑋

(𝑓, 𝑔 ∈ 𝑋) ,

(0.38)

i.e. we may take 𝑐 = 1. In this case, we call (𝑋, ‖⋅‖𝑋 ) a normalized algebra. An interesting question is if under the hypothesis (0.37), we can introduce an equivalent norm ⦀ ⋅ ⦀ such that the stronger condition (0.38) holds in the new norm. The following result shows how this may be achieved [192] for function spaces which are contained in the linear space 𝐵([𝑎, 𝑏]) of all bounded functions 𝑓 : [𝑎, 𝑏] → ℝ with norm¹⁷ (0.39)

‖𝑓‖∞ := sup |𝑓(𝑥)| . 𝑎≤𝑥≤𝑏

Proposition 0.31. Let (𝑋, ‖ ⋅ ‖𝑋 ) be a Banach algebra of functions 𝑓 : [𝑎, 𝑏] → ℝ which satisfies 𝑋 ⊆ 𝐵([𝑎, 𝑏]) and ‖𝑓𝑔‖𝑋 ≤ ‖𝑓‖∞ ‖𝑔‖𝑋 + ‖𝑓‖𝑋 ‖𝑔‖∞

(𝑓, 𝑔 ∈ 𝑋) ,

(0.40)

where ‖ ⋅ ‖∞ is defined by (0.39). Then the space 𝑋 equipped with the norm ⦀𝑓⦀𝑋 := ‖𝑓‖∞ + ‖𝑓‖𝑋

(𝑓 ∈ 𝑋)

(0.41)

is a normalized Banach algebra, i.e. ⦀𝑓𝑔⦀𝑋 ≤ ⦀𝑓⦀𝑋 ⦀𝑔⦀𝑋

(𝑓, 𝑔 ∈ 𝑋) .

(0.42)

Moreover, in case 𝑋 󳨅→ 𝐵([𝑎, 𝑏]), both norms ‖ ⋅ ‖𝑋 and ⦀ ⋅ ⦀𝑋 are equivalent. Proof. By the obvious estimate ‖𝑓𝑔‖∞ ≤ ‖𝑓‖∞ ‖𝑔‖∞ , for 𝑓, 𝑔 ∈ 𝑋, we get ⦀𝑓𝑔⦀𝑋 = ‖𝑓𝑔‖𝑋 + ‖𝑓𝑔‖∞ ≤ ‖𝑓𝑔‖𝑋 + ‖𝑓‖∞ ‖𝑔‖∞ ≤ ‖𝑓‖𝑋 ‖𝑔‖∞ + ‖𝑓‖∞ ‖𝑔‖𝑋 + ‖𝑓‖∞ ‖𝑔‖∞ ≤ ‖𝑓‖𝑋 ‖𝑔‖∞ + ‖𝑓‖∞ ‖𝑔‖𝑋 + ‖𝑓‖∞ ‖𝑔‖∞ + ‖𝑓‖𝑋 ‖𝑔‖𝑋 = (‖𝑓‖𝑋 + ‖𝑓‖∞ )(‖𝑔‖𝑋 + ‖𝑔‖∞ ) = ⦀𝑓⦀𝑋 ⦀𝑔⦀𝑋 ,

17 The norm (0.39) has to be carefully distinguished from the L∞ -norm (0.11). In fact, we may have ‖𝑓‖L∞ < ∞, but ‖𝑓‖∞ = ∞, if 𝑓 is unbounded on a nullset.

0.3 Basic function spaces | 25

which shows that (𝑋, ⦀ ⋅ ⦀𝑋 ) is a Banach algebra satisfying (0.42). If 𝑋 󳨅→ 𝐵([𝑎, 𝑏]) with sharp imbedding constant 𝑐(𝑋, 𝐵), then ⦀𝑓⦀𝑋 = ‖𝑓‖𝑋 + ‖𝑓‖∞ ≤ (1 + 𝑐(𝑋, 𝐵))‖𝑓‖𝑋 , while the converse estimate ‖𝑓‖𝑋 ≤ ⦀𝑓⦀𝑋 is of course trivial. An example of how to apply Proposition 0.31 will be given in the next section (Exam­ ple 0.43). The following is a useful method to generate from a given function space a whole sequence of new function spaces by considering derivatives. Definition 0.32. Given a space (𝑋, ‖ ⋅ ‖𝑋 ) of functions 𝑓 : [𝑎, 𝑏] → ℝ, we denote by 𝑋𝑛 (𝑛 ∈ ℕ) the set of all functions 𝑓 ∈ 𝑋 such that all derivatives 𝑓󸀠 , 𝑓󸀠󸀠 , . . . , 𝑓(𝑛) also belong to 𝑋. A natural norm on 𝑋𝑛 is then given by¹⁸ 𝑛−1

‖𝑓‖𝑋𝑛 := ∑ |𝑓(𝑗) (𝑎)| + ‖𝑓(𝑛) ‖𝑋 .

(0.43)

𝑗=0

A particularly important special case is 𝑛 = 1, i.e. the space of all 𝑓 for which the norm (0.44) ‖𝑓‖𝑋1 := |𝑓(𝑎)| + ‖𝑓󸀠 ‖𝑋 is finite.

◼

We will consider many examples of spaces of this type in Section 5.3 in connection with nonlinear operators. Let (𝑋, ‖⋅‖𝑋 ) and (𝑌, ‖⋅‖𝑌 ) be two function spaces such that 𝑋 󳨅→ 𝑌 with imbedding constant 𝑐(𝑋, 𝑌). If we associate to both 𝑋 and 𝑌 the corresponding spaces 𝑋𝑛 and 𝑌𝑛 with norm (0.43), for each 𝑓 ∈ 𝑋𝑛 , we have the estimate 𝑛−1

𝑛−1

‖𝑓‖𝑌𝑛 = ∑ |𝑓(𝑗) (𝑎)| + ‖𝑓(𝑛) ‖𝑌 ≤ ∑ |𝑓(𝑗) (𝑎)| + 𝑐‖𝑓(𝑛) ‖𝑋 ≤ max {𝑐, 1}‖𝑓‖𝑋𝑛 𝑗=0

𝑗=0

𝑛

𝑛

which shows that also 𝑋 󳨅→ 𝑌 . One may also show that the sharp imbedding con­ stant 𝑐(𝑋𝑛 , 𝑌𝑛 ) = max {𝑐(𝑋, 𝑌), 1} is independent of 𝑛.

0.3 Basic function spaces There are two important subspaces of the space (𝐵([𝑎, 𝑏]), ‖ ⋅ ‖∞ ) which will play a prominent role throughout this book. The first one is the classical Chebyshev space 𝐶([𝑎, 𝑏]) of all continuous functions 𝑓 : [𝑎, 𝑏] → ℝ with norm ‖𝑓‖𝐶 := max |𝑓(𝑥)| . 𝑎≤𝑥≤𝑏

18 As usual, 𝑓(0) := 𝑓.

(0.45)

26 | 0 Prerequisites It is well known that (𝐶([𝑎, 𝑏]), ‖ ⋅ ‖𝐶 ) is a Banach space, and the convergence with respect to the norm (0.45) coincides with the uniform convergence on [𝑎, 𝑏]. A typical problem in analysis and topology is that of extending a function from a small to a larger domain without “increasing its size.” One of the most important results of this type in functional analysis is the Hahn–Banach theorem for bounded linear functionals which we stated by Theorem 0.21. Another important extension re­ sult for continuous functions is the famous Tietze–Urysohn extension theorem which, in its simplest form, reads as follows. Theorem 0.33 (Tietze–Urysohn). Let 𝑀 ⊂ ℝ be closed and 𝑓 : 𝑀 → ℝ be continuous on 𝑀. Then there exists a continuous function 𝑓 ̂ : ℝ → ℝ such that ̂ sup {|𝑓(𝑥)| : 𝑥 ∈ ℝ} = sup {|𝑓(𝑥)| : 𝑥 ∈ 𝑀}

(0.46)

and 𝑓|̂ 𝑀 = 𝑓, i.e. 𝑓 ̂ extends 𝑓. We do not prove Theorem 0.33 as the proof of this result is quite technical and can be found in every topology textbook.¹⁹ It is clear that we need closedness of 𝑀 in The­ orem 0.33: if 𝑀 is not closed and 𝑥0 is an accumulation point of 𝑀 which does not belong to 𝑀, then the continuous function 𝑓(𝑥) := 1/(𝑥 − 𝑥0 ) does not have a contin­ uous extension to ℝ. We point out that Theorem 0.33 has an interesting consequence which is closely related to Luzin’s theorem (Theorem 0.2) for measurable functions. For a measurable set 𝑀 ⊆ ℝ and two functions 𝑓, 𝑔 : 𝑀 → ℝ, we set 𝛥(𝑓, 𝑔) = 𝛥(𝑓, 𝑔; 𝑀) := {𝑥 ∈ 𝑀 : 𝑓(𝑥) ≠ 𝑔(𝑥)} .

(0.47)

Then the following result follows by combining Luzin’s theorem and the Tiet­ ze–Urysohn extension theorem. Theorem 0.34 (Luzin). Let 𝑓 : [𝑎, 𝑏] → ℝ be a measurable function. Then for each 𝜀 > 0, we can find a continuous function 𝑓𝜀 : [𝑎, 𝑏] → ℝ such that 𝜆(𝛥(𝑓, 𝑓𝜀 )) < 𝜀 ,

(0.48)

where 𝛥(𝑓, 𝑓𝜀 ) is defined by (0.47). Apart from the space 𝐶([𝑎, 𝑏]), the second space we will often need in the sequel is the space 𝑅([𝑎, 𝑏]) of all regular functions 𝑓 ∈ 𝐵([𝑎, 𝑏]), i.e. bounded functions which have, at most, removable discontinuities or discontinuities of first kind (jumps). In

19 We point out that Theorem 0.33 holds not only on the real line, but in the much more general setting of metric spaces, and even for certain topological spaces, see, e.g. [320]. A certain variant of this theorem with a simple constructive proof may be found in Exercise 0.65.

0.3 Basic function spaces | 27

what follows, for 𝑓 : [𝑎, 𝑏] → ℝ, we use the notation 𝐷(𝑓) := {𝑥 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑓 is discontinuous at 𝑥} ,

(0.49)

𝐷0 (𝑓) := {𝑥 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑓 has a removable discontinuity at 𝑥} ,

(0.50)

𝐷1 (𝑓) := {𝑥 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑓 has a jump at 𝑥} .

(0.51)

and

So, we have (0.52)

𝐷(𝑓) = 𝐷0 (𝑓) ∪ 𝐷1 (𝑓) if 𝑓 is regular, and even 𝐷(𝑓) = 𝐷1 (𝑓) ,

𝐷0 (𝑓) = 0

(0.53)

if 𝑓 is monotone. Although a regular function may be very far from being monotone, it shares an important property with monotone functions: Proposition 0.35. The discontinuity set (0.49) of a regular function 𝑓 : [𝑎, 𝑏] → ℝ is at most countable. Proof. We adopt the usual notation lim 𝑓(𝑥) , 𝑓(𝑥0 −) := 𝑥→𝑥 0 𝑥𝑥0

(0.54)

for the unilateral limits of a function 𝑓 at some point 𝑥0 . So 𝑓 belongs to 𝑅([𝑎, 𝑏]) if and only if both limits (0.54) exist for all 𝑥0 ∈ (𝑎, 𝑏), as well as the unilateral limits 𝑓(𝑎+) and 𝑓(𝑏−). Given a regular function 𝑓 : [𝑎, 𝑏] → ℝ, we consider the “average function” 𝑓 of 𝑓 defined by²⁰ { 1 (𝑓(𝑥−) + 𝑓(𝑥+)) for 𝑥 ∈ 𝐷1 (𝑓) , 𝑓(𝑥) := { 2 𝑓(𝑥) otherwise . { Clearly, 𝐷(𝑓) = 𝐷(𝑓) , 𝐷0 (𝑓) = 𝐷0 (𝑓), 𝐷1 (𝑓) = 𝐷1 (𝑓) , (0.55) so it suffices to show that 𝐷(𝑓) is countable. Let 𝑥0 ∈ 𝐷0 (𝑓) ∩ (𝑎, 𝑏), and hence 𝑓(𝑥0 −) = 𝑓(𝑥0 +) ≠ 𝑓(𝑥0 ). Assume, for example, that 𝑓(𝑥0 −) = 𝑓(𝑥0 +) < 𝑓(𝑥0 ); then, 𝜀 := 12 (𝑓(𝑥0 ) − 𝑓(𝑥0 +)) > 0. Choose 𝛿 > 0 such that 𝑓(𝑥) < 𝑓(𝑥0 ) − 𝜀 for 𝑥 ∈ (𝑥0 − 𝛿, 𝑥0 ) ∪ (𝑥0 , 𝑥0 + 𝛿). This implies that the disc in ℝ2 centered at (𝑥0 , 𝑓(𝑥0 )) with radius min {𝛿, 𝜀} does not contain any other point of the graph of 𝑓 (or 𝑓) than (𝑥0 , 𝑓(𝑥0 )).

20 The average function 𝑓 is particularly useful in the theory of Fourier series; other kinds of regular­ ization will be used in Definitions 1.2 and 4.27.

28 | 0 Prerequisites Now, let 𝑥0 ∈ 𝐷1 (𝑓) ∩ (𝑎, 𝑏),and hence 𝑓(𝑥0 −) ≠ 𝑓(𝑥0 +). Assume, for example, that 𝑓(𝑥0 −) < 𝑓(𝑥0 +); then, 𝜀 := 13 (𝑓(𝑥0 +) − 𝑓(𝑥0 −)) > 0. Choose 𝛿 > 0 such that 𝑓(𝑥) < 𝑓(𝑥0 ) + 𝜀 for 𝑥 ∈ (𝑥0 − 𝛿, 𝑥0 ), and 𝑓(𝑥0 > 𝑓(𝑥0 +) − 𝜀 for 𝑥 ∈ (𝑥0 , 𝑥0 + 𝛿). This implies that the disc in ℝ2 centered at (𝑥0 , 𝑓(𝑥0 )) with radius min {𝛿, 𝜀} does not contain any other point of the graph of 𝑓 than (𝑥0 , 𝑓(𝑥0 )). Thus, we have shown that the set of all points on the graph of 𝑓, which are centers of the discs described above, contains only isolated points. By standard reasoning (see, e.g. [291]), this implies that the set of these points, and thus also the set 𝐷(𝑓), is at most countable. There is a remarkable result which links regular functions with continuous functions and goes back to Sierpiński [290]. Theorem 0.36 (Sierpiński). A function 𝑓 belongs to 𝑅([𝑎, 𝑏]) if and only if it can be rep­ resented as composition 𝑓 = 𝑔 ∘ 𝜏, where 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] is strictly increasing and 𝑔 ∈ 𝐶([𝑐, 𝑑]). Proof. We follow the idea of the proof in [290]. Suppose first that 𝑓 : [𝑎, 𝑏] → ℝ is reg­ ular. By Proposition 0.35, we then know that its discontinuity set (0.49) is countable, i.e. 𝐷(𝑓) = {𝑥1 , 𝑥2 , 𝑥3 , . . .} . We define a function 𝑝 : [𝑎, 𝑏] → ℝ by 𝑝(𝑥) := ∑ 𝑥>𝑥𝑛

1 , 2𝑛

(0.56)

where the sum is taken over all indices 𝑛 such that 𝑥𝑛 < 𝑥, and we put 𝑝(𝑥) := 0 if there are no such indices. Evidently, the function 𝑝 is increasing and takes its values in [0, 1]. Afterwards, we define 𝜏 : [𝑎, 𝑏] → ℝ by {𝑥 + 𝑝(𝑥) 𝜏(𝑥) := { 𝑥 + 𝑝(𝑥) + {

if 𝑥 ∈ ̸ 𝐷(𝑓) , 1 4𝑚

if 𝑥 = 𝑥𝑚 ∈ 𝐷(𝑓) .

(0.57)

Clearly, the function 𝜏 is strictly increasing, takes its values in [𝑐, 𝑑] = [𝑎, 𝑏 + 2], and satisfies 𝐷(𝜏) = 𝐷(𝑓). Being strictly monotone, the function 𝜏 admits an inverse 𝜏−1 : 𝐸 → [𝑎, 𝑏] on its range 𝐸 := 𝜏([𝑎, 𝑏]). So, we may define a map 𝑔 : 𝐸 → ℝ by 𝑔(𝑡) := 𝑓(𝜏−1 (𝑡))

(𝑡 ∈ 𝐸) .

(0.58)

By construction, this map satisfies 𝑓 = 𝑔∘𝜏 on [𝑎, 𝑏]; we claim that 𝑔 is continuous on the closure 𝐸 of 𝐸. First, we show that 𝑔 is continuous on 𝐸 = 𝜏([𝑎, 𝑏]). Thus, fix 𝑡∗ = 𝜏(𝑥∗ ) ∈ 𝐸. If 𝑥∗ ∈ 𝐷(𝑓), it is easy to see that 𝑡∗ is an isolated point of 𝐸, and there is nothing to prove. Therefore, assume that 𝑥∗ ∈ ̸ 𝐷(𝑓), which means that 𝑓 is continuous at 𝑥∗ .

0.3 Basic function spaces |

29

Given 𝜀 > 0, choose 𝛿 > 0 such that |𝑥 − 𝑥∗ | < 𝛿 implies |𝑓(𝑥) − 𝑓(𝑥∗ )| < 𝜀. Putting 𝑡1 := 𝜏(𝑥∗ − 𝛿) and 𝑡2 := 𝜏(𝑥∗ + 𝛿), we see that 𝑡1 < 𝑡∗ < 𝑡2 since 𝜏 is strictly increasing. Now, let 𝑡 be an arbitrary point in 𝐸 ∩ (𝑡1 , 𝑡2 ), and let 𝑥 := 𝜏−1 (𝑡). Then 𝜏(𝑥∗ − 𝛿) < 𝜏(𝑥) < 𝜏(𝑥∗ + 𝛿) , and hence 𝑥∗ − 𝛿 < 𝑥 < 𝑥∗ + 𝛿, i.e. |𝑥 − 𝑥∗ | < 𝛿, and so |𝑔(𝑡) − 𝑔(𝑡∗ )| = |𝑓(𝑥) − 𝑓(𝑥∗ )| < 𝜀 .

(0.59)

We have shown that for each 𝜀 > 0, we can find 𝑡1 , 𝑡2 such that 𝑡1 < 𝑡2 and (0.59) holds for any 𝑡 ∈ 𝐸 ∩ (𝑡1 , 𝑡2 ). However, this means that 𝑔 is continuous at 𝑡∗ , and also on the whole set 𝐸 since 𝑡∗ ∈ 𝐸 was arbitrary. One must now prove that 𝑔 is continuous at every accumulation point 𝑡∗ ∈ 𝐸 \ 𝐸. Being an accumulation point of 𝐸, the point 𝑡∗ may be approximated by sequences in 𝐸. However, we point out that we cannot find both an increasing sequence (𝑡𝑛 )𝑛 in 𝐸 and a decreasing sequence (𝑡󸀠𝑛 )𝑛 in 𝐸, both converging to 𝑡∗ . In fact, the elements 𝑥𝑛 := 𝜏−1 (𝑡𝑛 ) and 𝑥󸀠𝑛 := 𝜏−1 (𝑡󸀠𝑛 ) would then satisfy 𝑥𝑛 < 𝑥∗ < 𝑥󸀠𝑛 , where 𝑥𝑛 → 𝑥∗ as 𝑛 → ∞, and hence 𝑡𝑛 < 𝜏(𝑥∗ ) < 𝑡󸀠𝑛 . Letting 𝑛 → ∞ in this estimate, we get 𝑡∗ = 𝜏(𝑥∗ ), which means that 𝑡∗ ∈ 𝐸, contradicting our choice of 𝑡∗ . Therefore, we know that there exists either an increasing sequence (𝑡𝑛 )𝑛 in 𝐸 con­ verging to 𝑡∗ , or a decreasing sequence (𝑡󸀠𝑛 )𝑛 in 𝐸 converging to 𝑡∗ . Suppose that (𝑡𝑛 )𝑛 is increasing with 𝑡𝑛 → 𝑡∗ as 𝑛 → ∞. The sequence (𝑥𝑛 )𝑛 with 𝑥𝑛 := 𝜏−1 (𝑡𝑛 ) is then also increasing; moreover, (𝑥𝑛 )𝑛 is bounded from above because lim 𝜏(𝑥) = −∞,

𝑥→−∞

lim 𝜏(𝑥) = ∞ .

𝑥→∞

Thus, 𝑥𝑛 → 𝑥∗ for some 𝑥∗ ∈ [𝑎, 𝑏], and so lim 𝑓(𝑥𝑛 ) = lim 𝑔(𝜏(𝑥𝑛 )) = lim 𝑔(𝑡𝑛 ) .

𝑛→∞

𝑛→∞

𝑛→∞

Consequently, by putting lim 𝑔(𝑡𝑛 ) =: 𝑔(𝑡∗ ),

𝑛→∞

we see that the function 𝑔 is continuous, by construction, on the closure 𝐸 of 𝐸. Applying now Theorem 0.33 to 𝑀 := 𝐸, we get a continuous function 𝑔̂ : ℝ → ℝ which satisfies 𝑓 = 𝑔̂ ∘ 𝜏 on [𝑎, 𝑏]. This proves the “only if” part of our assertion. The proof of the “if part” is much simpler. Suppose that 𝑓 = 𝑔 ∘ 𝜏, where 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] is strictly increasing, and 𝑔 : [𝑐, 𝑑] → ℝ is continuous. Given 𝑥0 ∈ [𝑎, 𝑏], choose an increasing sequence (𝑥𝑛 )𝑛 in [𝑎, 𝑏] converging to 𝑥0 . Since 𝜏 is monotonically increasing, there exists 𝑡0 ∈ [𝑐, 𝑑] such that 𝜏(𝑥𝑛 ) → 𝑡0 as 𝑛 → ∞, and hence 𝑓(𝑥𝑛 ) = 𝑔(𝜏(𝑥𝑛 )) → 𝑔(𝑡0 )

(𝑛 → ∞)

by our continuity assumption on 𝑔. This shows that the left limit 𝑓(𝑥0 −) of 𝑓 at 𝑥0 exists. The existence of the right limit 𝑓(𝑥0 +) is proved similarly, and so we conclude that 𝑓 ∈ 𝑅([𝑎, 𝑏]).

30 | 0 Prerequisites In what follows, we will refer to Theorem 0.36 as the Sierpiński decomposition of 𝑓 ∈ 𝑅([𝑎, 𝑏]). Other types of decomposition will be considered in Theorems 1.28 and 1.41 in the next chapter. We illustrate Theorem 0.36 by means of a very simple example. Example 0.37. On [𝑎, 𝑏] = [0, 2], consider the function 𝑓 := 𝜒{1} . Clearly, 𝐷(𝑓) = 𝐷0 (𝑓) = {1}. We apply Sierpiński’s construction to this function to represent it in the form 𝑓 = 𝑔 ∘ 𝜏 with 𝑔 continuous and 𝜏 monotone. The functions (0.56) and (0.57) become {0 𝑝(𝑥) = { 1 {2

if 0 ≤ 𝑥 ≤ 1 , if 1 < 𝑥 ≤ 2

and 𝑡 { { { 𝜏(𝑡) = { 54 { { {𝑡 +

if 0 ≤ 𝑡 < 1 , if 𝑡 = 1 , 1 2

if 1 < 𝑡 ≤ 2 .

Therefore, 𝜏 is a strictly increasing bijection between [0, 2] and 𝐸 = [0, 1) ∪ { 45 } ∪ with inverse

( 32 , 52 ]

𝑠 { { { 𝜏 (𝑠) = {1 { { {𝑠 −

if 0 ≤ 𝑠 < 1 ,

−1

if 𝑠 = 1 2

if

3 2

5 4

,

0 such that |𝑓(𝑥) − 𝑓(𝑦)| ≤ 𝐿|𝑥 − 𝑦| (𝑎 ≤ 𝑥, 𝑦 ≤ 𝑏) .

(0.66)

More generally, 𝑓 is called Hölder continuous (or 𝛼-Lipschitz continuous for 0 < 𝛼 ≤ 1) if there exists a constant 𝐿 > 0 such that |𝑓(𝑥) − 𝑓(𝑦)| ≤ 𝐿|𝑥 − 𝑦|𝛼

(𝑎 ≤ 𝑥, 𝑦 ≤ 𝑏) .

(0.67)

We denote the set of all Lipschitz continuous functions on [𝑎, 𝑏] by Lip([𝑎, 𝑏]), and ◼ the set of all 𝛼-Lipschitz continuous functions on [𝑎, 𝑏] by Lip𝛼 ([𝑎, 𝑏]). Denoting by lip(𝑓) = lip(𝑓; [𝑎, 𝑏]) := sup 𝑥=𝑦 ̸

|𝑓(𝑥) − 𝑓(𝑦)| |𝑥 − 𝑦|

(0.68)

the minimal Lipschitz constant 𝐿 in (0.66) and, for 0 < 𝛼 < 1, by lip𝛼 (𝑓) = lip𝛼 (𝑓; [𝑎, 𝑏]) := sup 𝑥=𝑦 ̸

|𝑓(𝑥) − 𝑓(𝑦)| |𝑥 − 𝑦|𝛼

(0.69)

the minimal Hölder constant 𝐿 in (0.67), one may show that the spaces Lip([𝑎, 𝑏]) and Lip𝛼 ([𝑎, 𝑏]), equipped with the norms²¹ and

‖𝑓‖Lip := |𝑓(𝑎)| + lip(𝑓) ,

(0.70)

‖𝑓‖Lip𝛼 := |𝑓(𝑎)| + lip𝛼 (𝑓) ,

(0.71)

respectively, are Banach spaces. Moreover, one may easily prove that Lip𝛼 ([𝑎, 𝑏]) ⊆ Lip𝛽 ([𝑎, 𝑏]) (0 < 𝛽 ≤ 𝛼 ≤ 1) ,

21 Thus, in case 𝛼 = 1, we drop the subscript 1 and write lip and Lip rather than lip1 and Lip1 .

(0.72)

0.3 Basic function spaces | 33

where the inclusion is strict in case 𝛼 > 𝛽 (Exercise 0.45). The inclusions 𝐶1 ([𝑎, 𝑏]) ⊆ Lip([𝑎, 𝑏]) ⊆ Lip𝛼 ([𝑎, 𝑏]) ⊆ 𝐶([𝑎, 𝑏])

(0.73)

show that Lipschitz and Hölder continuity is some kind of “intermediate property” between continuity and continuous differentiability. The inclusions (0.72) and (0.73) are even continuous imbeddings: Proposition 0.40. For any 𝛼 ∈ (0, 1), the imbeddings 𝐶1 ([𝑎, 𝑏]) 󳨅→ Lip([𝑎, 𝑏]) 󳨅→ Lip𝛼 ([𝑎, 𝑏]) 󳨅→ 𝐶([𝑎, 𝑏])

(0.74)

hold. Proof. Let 𝐶1 ([𝑎, 𝑏]) be equipped with the first norm in (0.65), and let 𝑓 ∈ 𝐶1 ([𝑎, 𝑏]) and 𝑥, 𝑦 ∈ [𝑎, 𝑏] be fixed. By the mean value theorem, we then have |𝑓(𝑥) − 𝑓(𝑦)| = |𝑓󸀠 (𝜉)| |𝑥 − 𝑦| ≤ ‖𝑓󸀠 ‖𝐶 |𝑥 − 𝑦| for some 𝜉 between 𝑥 and 𝑦, and so lip(𝑓) ≤ ‖𝑓󸀠 ‖𝐶 . Comparing this with (0.68) yields ‖𝑓‖Lip = |𝑓(𝑎)| + lip(𝑓) ≤ ‖𝑓‖𝐶1

(𝑓 ∈ 𝐶1 ),

which means that 𝐶1 ([𝑎, 𝑏]) 󳨅→ Lip([𝑎, 𝑏]). Now, let 𝑓 ∈ Lip([𝑎, 𝑏]) and 𝐿 > lip(𝑓). Then |𝑓(𝑥) − 𝑓(𝑦)| |𝑓(𝑥) − 𝑓(𝑦)| |𝑥 − 𝑦|1−𝛼 ≤ 𝐿(𝑏 − 𝑎)1−𝛼 = |𝑥 − 𝑦|𝛼 |𝑥 − 𝑦| for 𝑥, 𝑦 ∈ [𝑎, 𝑏] and 𝛼 ∈ (0, 1). Since 𝐿 > lip(𝑓) was arbitrary, we conclude that Lip([𝑎, 𝑏]) 󳨅→ Lip𝛼 ([𝑎, 𝑏]). Finally, let 𝑓 ∈ Lip𝛼 ([𝑎, 𝑏]) and 𝐿 > lip𝛼 (𝑓). Then |𝑓(𝑥)| ≤ |𝑓(𝑥) − 𝑓(𝑎)| + |𝑓(𝑎)| ≤ 𝐿|𝑥 − 𝑎|𝛼 + |𝑓(𝑎)| ≤ 𝐿(𝑏 − 𝑎)𝛼 + |𝑓(𝑎)| . Since 𝐿 > lip𝛼 (𝑓) was arbitrary, we conclude that Lip𝛼 ([𝑎, 𝑏]) 󳨅→ 𝐶([𝑎, 𝑏]), and the proof is complete. The following simple example shows that all inclusions in (0.73) are strict for 𝛼 < 1. Example 0.41. Let [𝑎, 𝑏] = [0, 1]. The function 𝑓(𝑥) := |𝑥−𝑥0 |, where 𝑥0 ∈ (0, 1) is fixed, clearly belongs to Lip([0, 1]) with lip(𝑓) = 1, but is not differentiable at 𝑥0 . The function 𝑓(𝑥) = 𝑥𝛼 belongs to Lip𝛼 ([0, 1]), though in case 𝛼 < 1, not to Lip([0, 1]). Finally, let 𝑓 : [0, 1] → ℝ be defined by 1

{ 2 for 0 < 𝑥 ≤ 1 , (0.75) 𝑓(𝑥) := { log 𝑥 for 𝑥 = 0 . {0 L’Hospital’s rule shows that 𝑓 is continuous at zero, and thus on the whole inter­ val [0, 1]. On the other hand, for any 𝛼 > 0, we have lim

𝑥→0+

|𝑓(𝑥) − 𝑓(0)| 1 = lim 𝑥→0+ 𝑥𝛼 log 𝑥𝛼

and so 𝑓 ∈ ̸ Lip𝛼 ([0, 1]) for any 𝛼 ∈ (0, 1].

2 𝑥

= ∞, ♥

34 | 0 Prerequisites There is an analogous result²² to Theorem 0.33 for Hölder continuous (in particular, Lipschitz continuous) functions; here, the proof is even constructive. This result is usually referred to as McShane extension theorem [210]. Theorem 0.42 (McShane). Let 𝑀 ⊂ ℝ and 𝑓 : 𝑀 → ℝ be Hölder continuous on 𝑀 with Hölder exponent 𝛼 ∈ (0, 1]. Then there exists a Hölder continuous function 𝑓 ̂ : ℝ → ℝ such that 𝑓|̂ 𝑀 = 𝑓 and lip𝛼 (𝑓)̂ = lip𝛼 (𝑓). Proof. Define 𝑓 ̂ : ℝ → ℝ by ̂ := sup {𝑓(𝑧) − lip (𝑓)|𝑥 − 𝑧|𝛼 : 𝑧 ∈ 𝑀} (𝑥 ∈ ℝ) . 𝑓(𝑥) 𝛼

(0.76)

̂ Since lip𝛼 (𝑓)|𝑥 − 𝑧|𝛼 ≥ 0, with equality precisely for 𝑥 = 𝑧, it is clear that 𝑓(𝑥) = 𝑓(𝑥) for 𝑥 ∈ 𝑀. For general points 𝑥, 𝑦 ∈ ℝ, we have 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ̂ − 𝑓(𝑦)| ̂ |𝑓(𝑥) = 󵄨󵄨󵄨󵄨sup [𝑓(𝑧) − lip𝛼 (𝑓)|𝑥 − 𝑧|𝛼 ] − sup [𝑓(𝑧) − lip𝛼 (𝑓)|𝑦 − 𝑧|𝛼 ]󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨𝑧∈𝑀 𝑧∈𝑀 󵄨󵄨 󵄨󵄨 󵄨󵄨𝛼 𝛼 𝛼 󵄨󵄨 ≤ sup lip𝛼 (𝑓) 󵄨󵄨|𝑥 − 𝑧| − |𝑦 − 𝑧| 󵄨󵄨 ≤ lip𝛼 (𝑓) 󵄨󵄨𝑥 − 𝑦󵄨󵄨 . 𝑧∈𝑀

This shows that 𝑓 ̂ ∈ Lip𝛼 (ℝ) with lip𝛼 (𝑓)̂ ≤ lip𝛼 (𝑓). The converse inequality lip𝛼 (𝑓)̂ ≥ lip𝛼 (𝑓) is trivial since 𝑓 ̂ extends 𝑓. The Hölder spaces Lip𝛼 ([𝑎, 𝑏]) provide a good example for applying Proposition 0.31: Example 0.43. For 0 < 𝛼 ≤ 1, let 𝑋 = Lip𝑜𝛼 ([𝑎, 𝑏]) := {𝑓 ∈ Lip𝛼 ([𝑎, 𝑏]) : 𝑓(𝑎) = 0} be equipped with the the norm ‖𝑓‖Lip𝛼 = lip𝛼 (𝑓). Then 𝑋 ⊆ 𝐶([𝑎, 𝑏]) ⊆ 𝐵([𝑎, 𝑏]) and, for 𝑓, 𝑔 ∈ 𝑋 and 𝑥, 𝑦 ∈ [𝑎, 𝑏], lip𝛼 (𝑓𝑔) = sup 𝑥=𝑦 ̸

≤ sup 𝑥=𝑦 ̸

|(𝑓𝑔)(𝑥) − (𝑓𝑔)(𝑦)| |𝑥 − 𝑦|𝛼 |𝑓(𝑥)𝑔(𝑥) − 𝑓(𝑥)𝑔(𝑦)| |𝑓(𝑥)𝑔(𝑦) − 𝑓(𝑦)𝑔(𝑦)| + sup |𝑥 − 𝑦|𝛼 |𝑥 − 𝑦|𝛼 𝑥=𝑦 ̸

≤ sup |𝑓(𝑥)| sup 𝑎≤𝑥≤𝑏

𝑥=𝑦 ̸

|𝑔(𝑥) − 𝑔(𝑦)| |𝑓(𝑥) − 𝑓(𝑦)| + sup |𝑔(𝑦)| sup |𝑥 − 𝑦|𝛼 |𝑥 − 𝑦|𝛼 𝑥=𝑦 ̸ 𝑎≤𝑦≤𝑏

= ‖𝑓‖∞ lip𝛼 (𝑔) + ‖𝑔‖∞ lip𝛼 (𝑓) . This shows that (0.40) is satisfied since lip𝛼 (⋅) is a norm on 𝑋. By Proposition 0.31, we know that the norm ⦀𝑓⦀Lip𝛼 := ‖𝑓‖𝐶 + ‖𝑓‖𝑋 = max |𝑓(𝑥)| + lip𝛼 (𝑓) 𝑎≤𝑥≤𝑏

(𝑓 ∈ 𝑋)

(0.77)

22 As Theorem 0.33, the following Theorem 0.42 holds not only on the real line, but in the much more general setting of metric spaces.

0.3 Basic function spaces | 35

is equivalent to the norm (0.71) on Lip𝑜𝛼 ([𝑎, 𝑏]) and turns the space (Lip𝑜𝛼 , ⦀ ⋅ ⦀Lip𝛼 ) into a normalized Banach algebra. ♥ Of course, we may apply the construction of the higher order spaces 𝑋𝑛 also to 𝑋 = Lip([𝑎, 𝑏]) or, more generally, 𝑋 = Lip𝛼 ([𝑎, 𝑏]). This leads to the following Definition 0.44. According to (0.43), we equip the linear space Lip𝑛 ([𝑎, 𝑏]) with the natural norm 𝑛−1

‖𝑓‖Lip𝑛 := ∑ |𝑓(𝑗) (𝑎)| + ‖𝑓(𝑛) ‖Lip

(0.78)

𝑗=0

󸀠

(𝑛)

(𝑛)

= |𝑓(𝑎)| + |𝑓 (𝑎)| + . . . + |𝑓 (𝑎)| + lip(𝑓 ) or the equivalent norm 𝑛−1

⦀𝑓⦀Lip𝑛 := ∑ ‖𝑓(𝑗) ‖𝐶 + ‖𝑓(𝑛) ‖Lip 𝑗=0

(0.79)

= max |𝑓(𝑥)| + max |𝑓󸀠 (𝑥)| + . . . + max |𝑓(𝑛) (𝑥)| + lip(𝑓(𝑛) ) . 𝑎≤𝑥≤𝑏

𝑎≤𝑥≤𝑏

𝑎≤𝑥≤𝑏

The analogous construction for Hölder spaces is left to the reader (Exercise 0.46). ◼ In the study of both linear and nonlinear operators in function spaces, it is sometimes useful to know that restricting the discussion to a special domain (like [𝑎, 𝑏] = [0, 1]) does not affect the generality. To this end, we introduce some notation. Definition 0.45. Given real numbers 𝑎, 𝑏, 𝑐, 𝑑 with 𝑎 < 𝑏 and 𝑐 < 𝑑, consider the map ℓ : [𝑐, 𝑑] → [𝑎, 𝑏] defined by ℓ(𝑡) :=

𝑏−𝑎 (𝑡 − 𝑐) + 𝑎 (𝑐 ≤ 𝑡 ≤ 𝑑) . 𝑑−𝑐

(0.80)

Clearly, ℓ is an affine 𝐶∞ -diffeomorphism²³ between [𝑐, 𝑑] and [𝑎, 𝑏] with inverse ℓ−1 (𝑠) =

𝑑−𝑐 (𝑠 − 𝑎) + 𝑐 𝑏−𝑎

(𝑎 ≤ 𝑠 ≤ 𝑏) .

(0.81)

We call a function space 𝑋 shift-invariant if there exist numbers 𝑀, 𝑚 > 0 such that 𝑚‖𝑓‖𝑋([𝑎,𝑏]) ≤ ‖𝑓 ∘ ℓ‖𝑋([𝑐,𝑑]) ≤ 𝑀‖𝑓‖𝑋([𝑎,𝑏]) .

(0.82)

Here, the interval [𝑎, 𝑏] in 𝑋([𝑎, 𝑏]) denotes, of course, the domain of definition of functions from the space 𝑋. ◼

23 To be precise, we should write ℓ𝑎,𝑏,𝑐,𝑑 because (0.80) depends of course on our choice of intervals. However, we drop the indices so as not to overburden the notation.

36 | 0 Prerequisites The two-sided estimate (0.82) means that the linear operators 𝐿 and 𝐿−1 defined by 𝐿𝑓 = 𝑓 ∘ ℓ ,

𝐿−1 𝑔 = 𝑔 ∘ ℓ−1

(0.83)

are bounded from 𝑋([𝑎, 𝑏]) into 𝑋([𝑐, 𝑑]) respectively from 𝑋([𝑐, 𝑑]) into 𝑋([𝑎, 𝑏]). In some cases, one may even choose 𝑀 = 𝑚 = 1 in (0.82), which means that the operator 𝐿 in (0.83) is a linear surjective isometry. In this case, changing the interval does not increase or decrease the norm. We illustrate this by some simple examples. Example 0.46. A straightforward calculation shows that (0.82) holds with 𝑀 = 𝑚 = 1 for 𝑋 = 𝐶, 𝑋 = 𝐵, 𝑋 = L∞ , or 𝑋 = 𝑅. On the other hand, (0.82) holds for 𝑋 = L1 with 𝑚=𝑀=

𝑑−𝑐 , 𝑏−𝑎

and for 𝑋 = 𝐶1 with 𝑚 = min {

𝑏−𝑎 , 1} , 𝑑−𝑐

𝑀 = max {

𝑏−𝑎 , 1} . 𝑑−𝑐

(0.84)

To calculate the constants 𝑚 and 𝑀 for 𝑋 = Lip𝛼 , observe that lip𝛼 (𝑓 ∘ ℓ; [𝑐, 𝑑]) = (

𝑏−𝑎 𝛼 ) lip𝛼 (𝑓; [𝑎, 𝑏]) (0 < 𝛼 ≤ 1) . 𝑑−𝑐

Thus, we conclude that (0.82) holds for 𝑋 = Lip with 𝑚 and 𝑀 as in (0.84), and for 𝑋 = Lip𝛼 with 𝑚 = min {(

𝑏−𝑎 𝛼 ) , 1} , 𝑑−𝑐

𝑀 = max {(

𝑏−𝑎 𝛼 ) , 1} . 𝑑−𝑐

(0.85) ♥

Further calculations of 𝑚 and 𝑀 may be found in Exercises 0.57–0.60.

At this point, we introduce two interesting families of parameter-dependent functions which we will consider over and over in what follows to illustrate our abstract results. Definition 0.47. Given 𝛼, 𝛽 ∈ ℝ, consider the function 𝑓𝛼,𝛽 : [0, 1] → ℝ defined by {𝑥𝛼 sin 𝑥𝛽 𝑓𝛼,𝛽 (𝑥) := { 0 {

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 .

(0.86)

Clearly, this function is harmless for 𝛼 > 0 and 𝛽 > 0, but interesting for 𝛼 < 0 or 𝛽 < 0. We will call (0.86) the oscillatory function²⁴ determined by (𝛼, 𝛽) in what follows. ◼

24 Of course, the function (0.86) is “oscillatory” only for 𝛽 < 0; however, we use this name for all values of 𝛼 and 𝛽.

0.3 Basic function spaces |

37

It is very instructive to determine all values of 𝛼, 𝛽 ∈ ℝ for which 𝑓𝛼,𝛽 belongs to one of the spaces introduced so far. As a sample result, we do this for the spaces 𝐶, Lip, and 𝐶1 in the following Proposition 0.48. Further results in this direction may be found in Exercises 0.7, 0.8, 0.52, 0.54 and 0.55, and in many of the forthcoming chapters. Proposition 0.48. For 𝛼, 𝛽 ∈ ℝ, let 𝑓𝛼,𝛽 : [0, 1] → ℝ be defined by (0.86). Then the following holds. (a) 𝑓𝛼,𝛽 ∈ 𝐶([0, 1]) if and only if 𝛼 > 0 and 𝛽 is arbitrary, or 𝛼 ≤ 0 and 𝛽 > −𝛼. (b) 𝑓𝛼,𝛽 ∈ Lip([0, 1]) if and only if 𝛼 is arbitrary and 𝛽 ≥ 1 − 𝛼. (c) 𝑓𝛼,𝛽 ∈ 𝐶1 ([0, 1]) if and only if 𝛼 is arbitrary and 𝛽 > 1 − 𝛼. Proof. (a) The continuity of 𝑓𝛼,𝛽 in the case 𝛼 > 0 is clear since | sin 𝑥𝛽 | ≤ 1 for any 𝛽 ∈ ℝ. For 𝛼 = 0, we get the function 𝑓0,𝛽 (𝑥) = sin 𝑥𝛽 which is continuous at 0. Finally, in the case 𝛼 < 0, L’Hospital’s rule shows that lim

𝑥→0+

𝛽 sin 𝑥𝛽 = − lim 𝑥𝛼+𝛽 cos 𝑥𝛽 . 𝑥−𝛼 𝛼 𝑥→0+

(0.87)

So, in this case, 𝑓𝛼,𝛽 is continuous at 0 if and only if²⁵ 𝛽 > −𝛼. (b) Together with the function (0.86), we consider its “twin sister,” {𝑥𝛼 cos 𝑥𝛽 𝑔𝛼,𝛽 (𝑥) := { 0 {

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 .

(0.88)

Clearly, a continuous function 𝑓 : [𝑎, 𝑏] → ℝ which is differentiable on (𝑎, 𝑏) is Lipschitz continuous on [𝑎, 𝑏] if and only if its derivative is bounded on [𝑎, 𝑏]. This means that we have to find all pairs (𝛼, 𝛽) for which the derivative 󸀠 𝑓𝛼,𝛽 (𝑥) = 𝛼𝑓𝛼−1,𝛽 (𝑥) + 𝛽𝑔𝛼+𝛽−1,𝛽 (𝑥)

(0 < 𝑥 ≤ 1)

(0.89)

is bounded near zero. The second term in (0.89) is bounded near zero only if 𝛽 ≥ 1 − 𝛼. For 𝛼 ≥ 1, the first term is also bounded. On the other hand, for 𝛼 < 1, the factor 𝑥 󳨃→ 𝑥𝛼−1 is unbounded, but (0.89) shows that, nevertheless, the first term remains bounded if 𝛽 ≥ 1 − 𝛼. 󸀠 (0) for 𝑥 → 0+. In the (c) We have to show that the function (0.89) has the limit 𝑓𝛼,𝛽 case 𝛼 > 1 and 𝛽 > 1 − 𝛼, we have 󸀠 (𝑥) = 𝛼 lim 𝑓𝛼−1,𝛽 (𝑥) − 𝛽 lim 𝑔𝛼+𝛽−1,𝛽 (𝑥) = 0 lim 𝑓𝛼,𝛽

𝑥→0+

𝑥→0+

𝑥→0+

since both exponents 𝛼 − 1 and 𝛼 + 𝛽 − 1 are positive. Similarly, in the case 𝛼 = 1 and 𝛽 > 1 − 𝛼, the limit is zero since we still have 𝛽 > 0 in the first term. Finally, in the case

25 The limit (0.87) also exists in case 𝛽 = −𝛼, but has the “wrong” value 1, so 𝑓𝛼,−𝛼 has a removable discontinuity at zero.

38 | 0 Prerequisites 𝛼 < 1 and 𝛽 > 1 − 𝛼, L’Hospital’s rule shows that again 󸀠 lim 𝑓𝛼,𝛽 (𝑥) = 𝛼 lim 𝑓𝛼−1,𝛽 (𝑥) − 𝛽 lim 𝑔𝛼+𝛽−1,𝛽 (𝑥)

𝑥→0+

𝑥→0+

𝑥→0+

𝛼𝛽 =( − 𝛽) lim 𝑔𝛼+𝛽−1,𝛽 (𝑥) = 0 . 𝑥→0+ 1−𝛼 In all other cases, the limit does not exist. The second family of functions we will often use in the sequel is constructed over the interval [0, 1] as follows. Definition 0.49. Let 𝐶 = (𝑐𝑛 )𝑛 and 𝐷 = (𝑑𝑛 )𝑛 be two positive and decreasing real se­ quence converging to 0. In addition, (𝑐𝑛 )𝑛 is supposed to satisfy ∞

(0.90)

∑ 𝑐𝑛 = 1 .

𝑛=1

By means of such sequences, we construct a continuous function 𝑍𝐶,𝐷 : [0, 1] → ℝ in the following way. We put 𝑍𝐶,𝐷 (0) = 0 and let 𝑍𝐶,𝐷 increase linearly by 𝑑1 on the interval [0, 𝑐1 ] so that 𝑍𝐶,𝐷 (𝑐1 ) = 𝑑1 . Then we let 𝑍𝐶,𝐷 decrease linearly by 𝑑2 on [𝑐1 , 𝑐1 + 𝑐2 ], increase linearly by 𝑑3 on [𝑐1 + 𝑐2 , 𝑐1 + 𝑐2 + 𝑐3 ], decrease linearly by 𝑑4 on [𝑐1 + 𝑐2 + 𝑐3 , 𝑐1 + 𝑐2 + 𝑐3 + 𝑐4 ], and so on. So, the explicit form of 𝑍𝐶,𝐷 reads {0 { { 𝑍𝐶,𝐷 (𝑥) = {∑𝑛𝑘=1 (−1)𝑘+1 𝑑𝑘 { { {linear

for 𝑥 = 0 , for 𝑥 = ∑𝑛𝑘=1 𝑐𝑘 ,

(0.91)

otherwise .

We call (0.91) the (general) zigzag function determined by (𝐶, 𝐷) in what follows. A particularly important special case is 𝑐𝑛 :=

1 , 2𝑛

𝑑𝑛 :=

1 𝑛𝜃

(0.92)

for some 𝜃 > 0, i.e. 0 { { { 𝑍𝜃 (𝑥) = {1 − 21𝜃 + { { {linear

for 𝑥 = 0 , 1 3𝜃

− +... +

(−1)𝑛+1 𝑛𝜃

for 𝑥 = 1 −

1 2𝑛

,

(0.93)

otherwise .

In this case, we call (0.93) a special zigzag function.

◼

It follows from the construction and continuity of the zigzag function (0.91) that ∞

𝑍𝐶,𝐷 (1) = ∑ (−1)𝑘+1 𝑑𝑘 , 𝑘=1

∞

(−1)𝑘+1 . 𝑘𝜃 𝑘=1

𝑍𝜃 (1) = ∑

(0.94)

Again, it is illuminating to determine all sequences 𝐶 = (𝑐𝑛 )𝑛 and 𝐷 = (𝑑𝑛 )𝑛 for which the zigzag function (0.91) belongs to the function classes introduced so far. Of

0.3 Basic function spaces | 39

course, the function (0.91) is always continuous, by construction, but not differen­ tiable at its peaks. In the following proposition, we characterize all sequences 𝐶 = (𝑐𝑛 )𝑛 and 𝐷 = (𝑑𝑛 )𝑛 for which (0.91) is Hölder (in particular, Lipschitz) continuous. Proposition 0.50. For sequences 𝐶 = (𝑐𝑛 )𝑛 and 𝐷 = (𝑑𝑛 )𝑛 as above, the general zigzag function (0.91) belongs to Lip𝛼 ([0, 1]) (0 < 𝛼 ≤ 1) if and only if sup {𝑑𝑛 𝑐𝑛−𝛼 : 𝑛 = 1, 2, 3, . . .} < ∞ .

(0.95)

Proof. Denoting the supremum in (0.95) by 𝑆𝛼 (𝐶, 𝐷), we show that lip𝛼 (𝑍𝐶,𝐷 ; [0, 1]) ≤ 𝑆𝛼 (𝐶, 𝐷) ≤ 4 lip𝛼 (𝑍𝐶,𝐷 ; [0, 1]) .

(0.96)

The left inequality in (0.96) follows from a simple calculation, taking into account the slope of 𝑍𝐶,𝐷 between two successive peaks. To prove the right inequality, observe that the alternating series in (0.91) converges to some real number 𝑑∗ , say, since (𝑑𝑛 )𝑛 is monotonically decreasing and converges to zero. Now, if 𝑠 and 𝑡 > 𝑠 belong to the same interval between two adjacent peaks, we have |𝑍𝐶,𝐷 (𝑠) − 𝑍𝐶,𝐷 (𝑡)| ≤ 𝑆𝛼 (𝐶, 𝐷)|𝑠 − 𝑡|𝛼 , by definition of 𝑆𝛼 (𝐶, 𝐷). If 𝑠 and 𝑡 > 𝑠 belong to two intervals with just one peak in between, say 𝑠 < 𝑐1 + 𝑐2 + . . . + 𝑐𝑛 =: C𝑛 < 𝑡, then |𝑍𝐶,𝐷 (𝑠) − 𝑍𝐶,𝐷 (𝑡)| ≤ |𝑍𝐶,𝐷 (𝑠) − 𝑍𝐶,𝐷 (C𝑛 )| + |𝑍𝐶,𝐷 (C𝑛 ) − 𝑍𝐶,𝐷 (𝑡)| ≤ 𝑆𝛼 (𝐶, 𝐷)|𝑠 − C𝑛 |𝛼 + 𝑆𝛼 (𝐶, 𝐷)| C𝑛 −𝑡|𝛼 ≤ 2𝑆𝛼 (𝐶, 𝐷)|𝑠 − 𝑡|𝛼 . In general, let 𝑠 ̂ > 𝑠 be the smallest number at which 𝑍𝐶,𝐷 attains the value 𝑑∗ , and let 𝑡 ̂ < 𝑡 be the largest number with this property.²⁶ Then |𝑍𝐶,𝐷 (𝑠) − 𝑍𝐶,𝐷 (𝑡)| ̂ + |𝑍𝐶,𝐷 (𝑡)̂ − 𝑍𝐶,𝐷 (𝑡)| ̂ + |𝑍𝐶,𝐷 (𝑠)̂ − 𝑍𝐶,𝐷 (𝑡)| ≤ |𝑍𝐶,𝐷 (𝑠) − 𝑍𝐶,𝐷 (𝑠)| ̂ + |𝑍𝐶,𝐷 (𝑡)̂ − 𝑍𝐶,𝐷 (𝑡)| = |𝑍𝐶,𝐷 (𝑠) − 𝑍𝐶,𝐷 (𝑠)| ≤ 2𝑆𝛼 (𝐶, 𝐷)|𝑠 − 𝑠|̂ 𝛼 + 2𝑆𝛼 (𝐶, 𝐷)|𝑡 ̂ − 𝑡|𝛼 ≤ 4𝑆𝛼 (𝐶, 𝐷)|𝑠 − 𝑡|𝛼 . This proves the second estimate in (0.96), and we are done. In case of the special zigzag function (0.93), Proposition 0.50 implies the following somewhat surprising corollary. Corollary 0.51. The special zigzag function (0.93) does not belong to Lip𝛼 ([0, 1]) for any 𝜃 ∈ (0, 1] and 𝛼 ∈ (0, 1].

26 From a geometric reasoning, we see that 𝑠 ̂ belongs to the interval following 𝑠, and 𝑡 ̂ belongs to the interval preceding 𝑡.

40 | 0 Prerequisites Proof. Choosing 𝑐𝑛 and 𝑑𝑛 as in (0.92), we obtain sup {𝑑𝑛 𝑐𝑛−𝛼 : 𝑛 = 1, 2, 3, . . .} = sup {𝑛−𝜃 2𝑛𝛼 : 𝑛 = 1, 2, 3, . . .} = ∞ since the exponential term 2𝑛𝛼 grows essentially faster than the power type term 𝑛𝜃 . By (0.96), we see that lip𝛼 (𝑍𝜃 ; [0, 1]) = ∞, and so 𝑍𝜃 ∈ ̸ Lip𝛼 ([0, 1]) for any 𝛼 ∈ (0, 1]. The oscillation function (0.86) and the (special) zigzag function (0.93) are very useful for constructing counterexamples. For further reference, in the following Table 0.1, we collect the values of 𝛼, 𝛽, and 𝜃, respectively, for which these functions belong to the function spaces considered so far. An essential extension of this table will be given in Table 2.4 in Chapter 2. Table 0.1. Oscillation functions and zigzag functions.

belongs to 𝐶([0, 1]) iff belongs to 𝐶1 ([0, 1]) iff belongs to L1 ([0, 1]) iff belongs to Lip([0, 1]) iff belongs to Lip𝛾 ([0, 1]) iff

The function 𝑓𝛼,𝛽

The function 𝑍𝜃

𝛼 > 0 or 𝛼 ≤ 0 and 𝛼 + 𝛽 > 0 see Exercise 0.54 see Exercise 0.7 𝛼+𝛽≥1 see Exercise 0.52

always never always never never

It is clear that the zigzag function 𝑍𝜃 never belongs to spaces of differentiable func­ tions. On the other hand, being bounded on [0, 1], 𝑍𝜃 trivially belongs to L𝑝 ([0, 1]) for any 𝑝. Table 2.4 in Chapter 2 will contain other functions spaces in which the behavior of zigzag functions is more interesting. The Hölder spaces Lip𝛼 may be generalized in different ways; we consider one generalization which consists of replacing the map 𝑡 󳨃→ 𝑡𝛼 by a so-called modulus of continuity: Definition 0.52. An increasing function 𝜔 : [0, ∞) → [0, ∞) is called modulus of con­ tinuity if 𝜔(0) = 0 and 𝜔(𝑡) > 0 for 𝑡 > 0, 𝜔(𝑠 + 𝑡) ≤ 𝜔(𝑠) + 𝜔(𝑡), and 𝜔 is continuous at 0. ◼ A standard example of a modulus of continuity is of course 𝜔(𝑡) = 𝑡𝛼 for 0 < 𝛼 ≤ 1. Definition 0.52 is inspired by classical moduli of continuity which are defined for 𝑓 : [𝑎, 𝑏] → ℝ by 𝜔∞ (𝑓; 𝛿) = 𝜔∞ (𝑓, [𝑎, 𝑏]; 𝛿) := sup {|𝑓(𝑥 + ℎ) − 𝑓(𝑥)| : 𝑎 ≤ 𝑥 ≤ 𝑏 − ℎ}

(0.97)

0≤ℎ≤𝛿

and 𝑏−ℎ

{ } 𝜔𝑝 (𝑓; 𝛿) = 𝜔𝑝 (𝑓, [𝑎, 𝑏]; 𝛿) := sup { ∫ |𝑓(𝑥 + ℎ) − 𝑓(𝑥)|𝑝 𝑑𝑥} 0≤ℎ≤𝛿 {𝑎 } for 1 ≤ 𝑝 < ∞. In fact, the following is true.

1/𝑝

(0.98)

0.3 Basic function spaces |

41

Proposition 0.53. The relation (0.99)

lim 𝜔∞ (𝑓; 𝛿) = 0

𝛿→0+

holds for every function 𝑓 ∈ 𝐶([𝑎, 𝑏]), while the relation (0.100)

lim 𝜔𝑝 (𝑓; 𝛿) = 0

𝛿→0+

holds for every 𝑓 ∈ L𝑝 ([𝑎, 𝑏]). Proof. The validity of (0.99) for continuous 𝑓 is clear. To prove that (0.100) holds for 𝑓 ∈ L𝑝 ([𝑎, 𝑏]), we fix 𝜀 > 0 and choose a continuous function²⁷ 𝑔 : [𝑎, 𝑏] → ℝ such that ‖𝑓 − 𝑔‖L𝑝 ≤ 𝜀. Since 𝑔 is uniformly continuous on [𝑎, 𝑏], we find a 𝛿 > 0 such that |𝑔(𝑥) − 𝑔(𝑦)| ≤ 𝜀(𝑏 − 𝑎)−1/𝑝 for 𝑥, 𝑦 ∈ [𝑎, 𝑏] satisfying |𝑥 − 𝑦| ≤ 𝛿. Combining these two estimates, for 0 ≤ ℎ ≤ 𝛿, we get 1/𝑝

𝑏−ℎ

{ } 𝑝 { ∫ |𝑓(𝑥 + ℎ) − 𝑓(𝑥)| 𝑑𝑥} {𝑎 } 1/𝑝

𝑏−ℎ

≤ 2‖𝑓 − 𝑔‖L𝑝 ≤ 2𝜀 +

{ } + { ∫ |𝑔(𝑥 + ℎ) − 𝑔(𝑥)| 𝑑𝑥} {𝑎 }

𝜀 (𝑏 − 𝑎)1/𝑝 = 3𝜀, (𝑏 − 𝑎)1/𝑝

proving the assertion. Proposition 0.53 states that 𝜔𝑝 (𝑓; 𝛿) = 𝑜(1)

(𝛿 → 0+)

for every 𝑓 ∈ L𝑝 . One may show that the essentially stronger condition 𝜔𝑝 (𝑓; 𝛿) = 𝑜(𝛿)

(𝛿 → 0+)

is satisfied only for constant functions, see Exercise 0.76. By means of the moduli (0.97) and (0.98), we may define generalized Hölder spaces as follows. Definition 0.54. Let 𝜔 : [0, ∞) → [0, ∞) be an arbitrary modulus of continuity in the sense of Definition 0.52, and let 𝜔∞ (𝑓; ⋅) and 𝜔𝑝 (𝑓; ⋅) be defined by (0.97) and (0.98), respectively. For 𝑓 : [𝑎, 𝑏] → ℝ, put lip𝜔,𝑝 (𝑓) = lip𝜔,𝑝 (𝑓; [𝑎, 𝑏]) := sup 𝛿>0

𝜔𝑝 (𝑓; 𝛿) 𝜔(𝛿)

(1 ≤ 𝑝 ≤ ∞) .

(0.101)

27 Here, we use the fact that the space 𝐶([𝑎, 𝑏]) is dense in the space L𝑝 ([𝑎, 𝑏]) with respect to the norm (0.11).

42 | 0 Prerequisites We write 𝑓 ∈ Lip𝜔,𝑝 ([𝑎, 𝑏]) if lip𝜔,𝑝 (𝑓) < ∞ and call Lip𝜔,𝑝 ([𝑎, 𝑏]) the generalized Hölder space defined by 𝜔 and 𝑝. We consider this space equipped with the norm {‖𝑓‖𝐶 + lip𝜔,∞ (𝑓) ‖𝑓‖Lip𝜔,𝑝 := { ‖𝑓‖ + lip𝜔,𝑝 (𝑓) { L𝑝

if 𝑝 = ∞ , if 𝑝 < ∞ .

(0.102)

In case 𝜔(𝑡) = 𝑡𝛼 (0 < 𝛼 < 1), we write lip𝛼,𝑝 (𝑓) instead of lip𝜔,𝑝 (𝑓) and Lip𝛼,𝑝 ([𝑎, 𝑏]) instead of Lip𝜔,𝑝 ([𝑎, 𝑏]). ◼ Clearly, taking 𝑝 = ∞ in Definition 0.54, we get the Hölder space Lip𝛼,∞ ([𝑎, 𝑏]) = Lip𝛼 ([𝑎, 𝑏]); this follows from the fact that the condition 𝜔∞ (𝑓, 𝛿) ≤ 𝜔(𝛿) for all 𝛿 ≥ 0 is equivalent to the condition |𝑓(𝑠) − 𝑓(𝑡)| ≤ 𝜔(|𝑠 − 𝑡|) (𝑎 ≤ 𝑠, 𝑡 ≤ 𝑏) . We remark that the spaces Lip𝛼,𝑝 are related to the classical Hölder spaces Lip𝛽 through the imbeddings Lip𝛼 ([𝑎, 𝑏]) 󳨅→ Lip𝛼,𝑝 ([𝑎, 𝑏]) 󳨅→ Lip𝛼−1/𝑝 ([𝑎, 𝑏]) (Exercise 0.71). We will return to this class of spaces in Chapter 2, where we will compare them with imbeddings into spaces of functions of generalized bounded variation. To conclude this section on function spaces, we briefly consider compact sets. In many applications, it is important to describe compact subsets of a Banach space by an “intrinsic” characterization. The simplest such description is known in the finite dimensional space ℝ𝑛 , where a subset 𝐾 ⊂ ℝ𝑛 is compact if and only if 𝐾 is both closed and bounded.²⁸ On the other hand, in every infinite dimensional space, closedness and boundedness is still necessary for compactness, but never sufficient. In fact, in such spaces, one has to add a third property (called “compactness criterion”) to get compactness. We briefly recall two well-known compactness criteria, the first in the space 𝐶([𝑎, 𝑏]), and the second in the space L𝑝 ([𝑎, 𝑏]) for 1 ≤ 𝑝 < ∞. They both build on the moduli of continuity (0.97) and (0.98); we cite them without proof. Proposition 0.55 (Arzelà–Ascoli). Let 𝜔∞ (𝑓; 𝛿) be defined by (0.97). Then a subset 𝐾 ⊂ 𝐶([𝑎, 𝑏]) is compact if and only if 𝐾 is closed, bounded, and satisfies (0.99) uniformly with respect to 𝑓 ∈ 𝐾, i.e. lim sup {𝜔∞ (𝑓; 𝛿) : 𝑓 ∈ 𝐾} = 0 .

𝛿→0+

28 This result is known as Heine–Borel compactness criterion.

(0.103)

0.4 Comments on Chapter 0 | 43

Proposition 0.56 (Kolmogorov). Let 𝜔𝑝 (𝑓; 𝛿) be defined by (0.99). Then a subset 𝐾 ⊂ L𝑝 ([𝑎, 𝑏]) is compact if and only if 𝐾 is closed, bounded, and satisfies (0.100) uniformly with respect to 𝑓 ∈ 𝐾, i.e. lim sup {𝜔𝑝 (𝑓; 𝛿) : 𝑓 ∈ 𝐾} = 0 .

𝛿→0+

(0.104)

The property expressed in relation (0.103) is usually called the equicontinuity of 𝐾. Therefore, if (𝑓𝑛 )𝑛 is a bounded sequence in an equicontinuous set 𝐾 ⊂ 𝐶([𝑎, 𝑏]), we may always find a subsequence which is uniformly convergent on [𝑎, 𝑏]. A weak ana­ logue to this in the space of functions of bounded variation will be proved in Theo­ rem 1.11 in the first chapter.

0.4 Comments on Chapter 0 In any textbook on measure and integration theory, the Lebesgue spaces L𝑝 constitute an important ingredient. In our examples and counterexamples in Section 0.1, we fol­ lowed the book [39] which is an extremely valuable source. Classical monographs on Orlicz spaces are [169] or [263]. The general theorems on Banach spaces and bounded linear functionals treated in Section 0.2 may be found in every textbook on functional analysis, we mention [146, 171, 266, 270, 302, 319, 320]. A standard reference on function spaces is [172]. An updated version [249] of it just appeared in this book series. All of function spaces we have discussed in this chapter are Banach spaces, and the same is true for other spaces we will introduce in the following chapters. An ex­ ample of an incomplete function space is the linear space 𝑃([𝑎, 𝑏]) of polynomials 𝑝(𝑥) = 𝑎𝑛 𝑥𝑛 + 𝑎2 𝑥2 + 𝑎1 𝑥 + 𝑎0 (of any degree 𝑛) with the norm ‖𝑝‖𝐶 = max |𝑝(𝑥)| 𝑎≤𝑥≤𝑏

(0.105)

inherited from the larger space 𝐶([𝑎, 𝑏]). Since 𝑃([𝑎, 𝑏]) is not closed²⁹ in 𝐶([𝑎, 𝑏]), the space (𝑃([𝑎, 𝑏]), ‖ ⋅ ‖𝐶 ) is incomplete. For 𝑚(𝑓) as in (0.61) and 𝑀(𝑓) as in (0.62), in 1925, Banach [40] introduced the indicatrix 𝐼𝑓 : [𝑚(𝑓), 𝑀(𝑓)] → [0, ∞] of a continuous function 𝑓 : [𝑎, 𝑏] → ℝ as 𝐼𝑓 (𝑦) := #{𝑓−1 (𝑦) ∩ [𝑎, 𝑏]} ,

(0.106)

29 In fact, the classical Weierstrass approximation theorem shows that the closure of 𝑃([𝑎, 𝑏]) in the norm (0.45) is the whole space 𝐶([𝑎, 𝑏]). Even worse, since the monomials form a countably infinite (algebraic) basis in 𝑃([𝑎, 𝑏]), it follows from the so-called Baire category theorem that there is no norm on 𝑃([𝑎, 𝑏]) which makes this space complete, see also Exercise 0.27.

44 | 0 Prerequisites where #𝐴 denotes the cardinality of the set 𝐴. He also proved that 𝑓 has bounded variation if and only if 𝑀(𝑓)

(0.107)

∫ 𝐼𝑓 (𝑦) 𝑑𝑦 < ∞ , 𝑚(𝑓)

see Proposition 1.27 in the next chapter. Later, Natanson [238] and others called 𝑁𝑓 the Banach indicatrix of 𝑓. If 𝑓 is merely regular, one may use the Sierpiński decom­ position ( Theorem 0.36) 𝑓 = 𝑔 ∘ 𝜏 of 𝑓, with 𝑔 being continuous und 𝜏 being strictly increasing, to extend Banach’s definition to 𝑓 ∈ 𝑅([𝑎, 𝑏]) by putting 𝑁𝑓 := 𝑁𝑔 . This ex­ tension due to Lozinskij [186] is possible since a strictly increasing change of variables does not change the number of elements in 𝑓−1 (𝑦). Some authors call the estimate (0.16) Hölder inequality. However, usually this term refers to the inequality 𝑛

𝑛

∑ 𝛼𝑘 𝛽𝑘 ≤ ( ∑

𝑘=1

𝑘=1

1/𝑝 𝑝 𝛼𝑘 )

𝑛

(∑ 𝑘=1

1/𝑝󸀠 𝑝󸀠 𝛽𝑘 )

,

(0.108)

where 𝛼1 , . . . , 𝛼𝑛 , 𝛽1 , . . . , 𝛽𝑛 are positive real numbers and the exponents 𝑝 and 𝑝󸀠 are related by (0.13). Regular functions (also called quasicontinuous functions by some authors, e.g. [76]), are closely related to step functions. Recall that 𝑓 : [𝑎, 𝑏] → ℝ is called a step function if finitely many points 𝑎 = 𝑡0 < 𝑡1 < . . . < 𝑡𝑚 = 𝑏 exist such that 𝑓 is constant³⁰ on each open interval (𝑡𝑗−1 , 𝑡𝑗 ) for 𝑗 = 1, 2, . . . , 𝑚. We will write 𝑆([𝑎, 𝑏]) for the linear space of all step functions on [𝑎, 𝑏]. Obviously, the (strict) inclusion 𝑆([𝑎, 𝑏]) ⊂ 𝑅([𝑎, 𝑏]) holds, which means that step functions are regular. The following proposition which we will prove in view of its importance makes this more precise. Proposition 0.57. The equality 𝑆([𝑎, 𝑏]) = 𝑅([𝑎, 𝑏])

(0.109)

holds, where the closure in (0.109) is taken in the norm (0.39). Proof. First, suppose that 𝑓 ∈ 𝑆([𝑎, 𝑏]), which means that for every 𝜀 > 0, there exists a step function 𝑔 such that ‖𝑓 − 𝑔‖∞ ≤ 𝜀. We have to show that 𝑓 has unilateral limits at every point. Fix 𝑥0 ∈ [𝑎, 𝑏); we show that the right limit 𝑓(𝑥0 +) exists, see (0.54). Since 𝑔 is a step function, we can find a 𝛿 > 0 such that 𝑔 is constant on (𝑥0 , 𝑥0 + 𝛿). However, this implies that |𝑓(𝑠) − 𝑓(𝑡)| ≤ |𝑓(𝑠) − 𝑔(𝑠)| + |𝑔(𝑠) − 𝑔(𝑡)| + |𝑔(𝑡) − 𝑓(𝑡)| ≤ 2𝜀 for 𝑠, 𝑡 ∈ (𝑥0 , 𝑥0 + 𝛿), and so 𝑓(𝑥0 +) exists. The proof for the left limit 𝑓(𝑥0 −) for 𝑥0 ∈ (𝑎, 𝑏] is similar, and so we have shown that 𝑓 is regular.

30 Here, 𝑓 is allowed to take on any arbitrary value at the points 𝑡0 , 𝑡1 , . . . , 𝑡𝑚 themselves.

0.4 Comments on Chapter 0 | 45

Conversely, suppose now that 𝑓 ∈ 𝑅([𝑎, 𝑏]), and let 𝜀 > 0. For each 𝑥 ∈ [𝑎, 𝑏], we can find a 𝛿 = 𝛿(𝜀, 𝑥) such that |𝑓(𝑠)−𝑓(𝑡)| ≤ 𝜀 for 𝑠, 𝑡 ∈ (𝑥−𝛿(𝜀, 𝑥), 𝑥)∪(𝑥, 𝑥+𝛿(𝜀, 𝑥)), by definition of the unilateral limits (0.54). Since the intervals {((𝑥 − 𝛿(𝜀, 𝑥), 𝑥 + 𝛿(𝜀, 𝑥)) : 𝑎 ≤ 𝑥 ≤ 𝑏} form an open cover and [𝑎, 𝑏] is compact, we only actually need finitely many of them to do the job, say [𝑎, 𝑏] ⊆ (𝑥0 − 𝛿(𝜀, 𝑥0 ), 𝑥0 + 𝛿(𝜀, 𝑥0 )) ∪ . . . ∪ (𝑥𝑛 − 𝛿(𝜀, 𝑥𝑛 ), 𝑥𝑛 + 𝛿(𝜀, 𝑥𝑛 )) . Choose points 𝑎 = 𝑡0 < 𝑡1 < . . . < 𝑡𝑚 = 𝑏 such that each interval (𝑡𝑗−1 , 𝑡𝑗 ) is contained in some (𝑥𝑖 − 𝛿(𝜀, 𝑥𝑖 ), 𝑥𝑖 ) or in some (𝑥𝑖 , 𝑥𝑖 + 𝛿(𝜀, 𝑥𝑖 )). Then |𝑓(𝑠) − 𝑓(𝑡)| ≤ 𝜀 whenever 𝑠, 𝑡 ∈ (𝑡𝑗−1 , 𝑡𝑗 ). Now, define 𝑔 : [𝑎, 𝑏] → ℝ by {𝑓(𝑡𝑗 ) for 𝑥 = 𝑡𝑗 (𝑗 = 0, 1, . . . , 𝑚) , 𝑔(𝑥) := { 1 𝑓 ( (𝑡 + 𝑡 ) for 𝑡𝑗−1 < 𝑥 < 𝑡𝑗 (𝑗 = 1, 2, . . . , 𝑚) . { 2 𝑗 𝑗−1 Clearly, 𝑔 ∈ 𝑆([𝑎, 𝑏]) and ‖𝑓 − 𝑔‖ ≤ 𝜀, so 𝑓 ∈ 𝑆([𝑎, 𝑏]). The shift invariance of some spaces in the sense of Definition 0.45 is discussed in [16] in connection with nonlinear composition operators; we will come back to this in Chap­ ter 5. The oscillation functions are studied in detail in [11], while the zigzag functions are discussed in [250] in connection with certain spaces of functions of bounded vari­ ation which we will study in Section 2.2. Compactness criteria like those given in Propositions 0.55 and 0.56 are extremely useful for proving existence theorems in both linear and nonlinear functional analy­ sis. The range of applications of such existence theorems may be enlarged by consid­ ering so-called measures of noncompactness which are closely related to the charac­ teristics (0.103) and (0.104), see, e.g. [6, 10, 26, 41, 42]. The generalized Hölder spaces Lip𝜔,𝑝 and Lip𝛼,𝑝 introduced in Definition 0.54 are quite useful in the theory of Fourier series which we will briefly discuss in Chapter 7. In particular, it is interesting to establish continuous imbeddings between such spaces. For example, the following two results [141, 142, 305] are known: Proposition 0.58 (Hardy–Littlewood). Let 𝜔 : [0, ∞) → [0, ∞) be a modulus of conti­ nuity, and let 0 < 𝛽 ≤ 1 ≤ 𝑝 < 𝑞 < ∞. Then the imbedding Lip𝜔,𝑝 ([𝑎, 𝑏]) 󳨅→ Lip𝛽,𝑞 ([𝑎, 𝑏]) holds if and only if 𝜔(𝛿) < ∞. 𝛿𝛽+1/𝑝−1/𝑞 In particular, Lip𝛼,𝑝 ([𝑎, 𝑏]) 󳨅→ Lip𝛽,𝑞 ([𝑎, 𝑏]) if and only if lim

𝛿→0+

𝛼−𝛽≥ for 𝛽 < 𝛼 ≤ 1.

1 1 − 𝑝 𝑞

46 | 0 Prerequisites Proposition 0.59 (Ulyanov). Let 𝜔 : [0, ∞) → [0, ∞) be a modulus of continuity, and let 1 ≤ 𝑝 < 𝑞 < ∞. Then the imbedding Lip𝜔,𝑝 ([𝑎, 𝑏]) 󳨅→ L𝑞 ([𝑎, 𝑏]) holds if and only if

∞ 1 ∑ 𝑛𝑞/𝑝−2 𝜔𝑞 ( ) < ∞ . 𝑛 𝑛=1

In particular, Lip𝛼,𝑝 ([𝑎, 𝑏]) 󳨅→ L𝑞 ([𝑎, 𝑏]) if and only if 𝑝 − 𝛼𝑞 < 1 𝑞 for 0 < 𝛼 ≤ 1. More imbedding theorems of this type, also in connection with functions of bounded variation, will be discussed in Sections 2.2 and 2.8.

0.5 Exercises to Chapter 0 We state some exercises on the topics covered in this chapter; exercises marked with an asterisk ∗ are more difficult. Exercise 0.1. Let 𝑀 ⊆ ℝ be a measurable set and 𝑓 : 𝑀 → ℝ a measurable function. For 𝑐 > 0, let 𝑀𝑐 (𝑓) := {𝑥 ∈ 𝑀 : |𝑓(𝑥)| > 𝑐} . Prove that inf {𝑐 > 0 : 𝜆(𝑀𝑐 (𝑓)) = 0} = inf sup {|𝑔(𝑥)| : 𝑥 ∈ 𝑀} , 𝑓∼𝑔

where the supremum is taken over all functions 𝑔 : 𝑀 → ℝ which are equivalent to 𝑓. Exercise 0.2. From the proof of Proposition 0.10 (b), it follows that L∞ ([𝑎, 𝑏]) 󳨅→ L𝑞 ([𝑎, 𝑏]) 󳨅→ L𝑝 ([𝑎, 𝑏]) 󳨅→ L1 ([𝑎, 𝑏])

(1 < 𝑝 < 𝑞 < ∞) .

Calculate the sharp imbedding constants 𝑐(L∞ , L𝑞 ), 𝑐(L𝑞 , L𝑝 ), and 𝑐(L𝑝 , L1 ) for these values of 𝑝 and 𝑞. Exercise 0.3. Let 1 ≤ 𝑝 < 𝑞 ≤ ∞, and suppose that 𝑓 ∈ L𝑝 (𝐼) ∩ L𝑞 (𝐼), where 𝐼 ⊆ ℝ is an unbounded interval. Show that then 𝑓 ∈ ⋂ L𝑟 (𝐼) . 𝑝≤𝑟≤𝑞

Why is this statement trivial for bounded intervals 𝐼?

0.5 Exercises to Chapter 0 |

47

Exercise 0.4. The convolution 𝑓 ∗ 𝑔 of two functions 𝑓, 𝑔 ∈ L1 (ℝ) is defined by ∞

(𝑓 ∗ 𝑔)(𝑥) := ∫ 𝑓(𝑥 − 𝑡)𝑔(𝑡) 𝑑𝑡 . −∞

Use Fubini’s theorem (Theorem 0.7) for 𝑝 = 𝑞 = 1 to show that (L1 (ℝ), ∗) is a Banach algebra such that ‖𝑓 ∗ 𝑔‖L1 ≤ ‖𝑓‖L1 ‖𝑔‖L1 for all 𝑓, 𝑔 ∈ L1 (ℝ). Calculate the convolution 𝑓 ∗ 𝑓 of the characteristic function 𝑓 = 𝜒[𝑎,𝑏] with itself. Exercise 0.5. Let 𝜌 : ℝ → ℝ be defined by 1 {𝑐 exp (− 1−|𝑥| for |𝑥| < 1 , 2) 𝜌(𝑥) := { 0 for |𝑥| ≥ 1 , {

where

∞

𝑐 := ∫ 𝜌(𝑥) 𝑑𝑥 . −∞

For 𝜀 > 0, define 𝜌𝜀 : ℝ → ℝ by 𝜌𝜀 (𝑥) :=

𝑥 1 𝜌( ) . 𝜀 𝜀

The function 𝜌𝜖 is called the mollifier; this name is explained by the fact that 𝜌𝜀 ∗ 𝑓 ∈ 𝐶∞ (ℝ) for every 𝑓 ∈ L1 (ℝ), where ∗ denotes the convolution introduced in Exer­ cise 0.4. Prove this and calculate the derivative of 𝜌𝜀 ∗ 𝑓. Exercise 0.6*. With 𝜌 and 𝜌𝜀 as in Exercise 0.5, prove that ‖𝜌𝜀 ∗ 𝑓‖L1 ≤ ‖𝑓‖L1

(𝜀 > 0)

and lim ‖𝜌𝜀 ∗ 𝑓 − 𝑓‖L1 = 0

𝜀→0+

for every 𝑓 ∈ L1 (ℝ). Conclude that 𝑓 ∈ L1 (ℝ) is separable. Exercise 0.7. Show that the function 𝑓𝛼,𝛽 from (0.86) belongs to L1 ([0, 1]) if and only if 𝛽 ≥ 0 and 𝛼 + 𝛽 > −1, or 𝛽 < 0 and 𝛼 > −1. Exercise 0.8. Show that the function 𝑓𝛼,𝛽 from (0.86) belongs to L1 ([1, ∞)) if and only if 𝛽 ≤ 0 and 𝛼 + 𝛽 < −1, or 𝛽 > 0 and 𝛼 < −1. Exercise 0.9. For 𝜇 ∈ ℝ, and let 𝑓𝜇 : ℝ → ℝ be defined by 𝜇

{ | log1/𝑝𝑥| for 𝑥 ≠ 0 , 𝑓𝜇 (𝑥) := { 𝑥 0 for 𝑥 = 0 . { Prove that 𝑓𝜇 ∈ L𝑝 ([0, 1]) if and only if 𝜇𝑝 < −1.

48 | 0 Prerequisites Exercise 0.10. Determine all values of 𝜇 for which the functions 𝑓𝜇 from Exercise 0.9 belongs to L𝑝 ([1, ∞)). Exercise 0.11. Let 𝑀 ⊆ ℝ be a measurable set. A sequence (𝑓𝑛 )𝑛 of measurable func­ tions 𝑓𝑛 : 𝑀 → ℝ converges in measure to some measurable function 𝑓 : 𝑀 → ℝ if lim 𝜆(𝑀𝑐 (𝑓𝑛 − 𝑓)) = 0 𝑛→∞

for all 𝑐 > 0, where 𝑀𝑐 (𝑓) is defined as in Exercise 0.1. Show that convergence of (𝑓𝑛 )𝑛 to 𝑓 a.e. on 𝑀 implies convergence of (𝑓𝑛 )𝑛 in measure to 𝑓, though not vice versa. Exercise 0.12. Let 𝜙 : [0, ∞) → [0, ∞) be an increasing continuous function satisfy­ ing 𝜙(0) = 0 and 𝜙(𝑢) > 0 for 𝑢 > 0. Let 𝑀 ⊆ ℝ be a measurable set, and let 𝑓𝑛 : 𝑀 → ℝ and 𝑓 : 𝑀 → ℝ be measurable functions such that ∫ 𝜙(|𝑓𝑛 (𝑥) − 𝑓(𝑥)|) 𝑑𝑥 → 0 (𝑛 → ∞) . 𝑀

Prove that the sequence (𝑓𝑛 )𝑛 converges in measure on 𝑀 to 𝑓 (Exercise 0.11). Exercise 0.13. For a function 𝑓 ∈ L1 ([𝑎, 𝑏]), prove the equality 𝑏

∞

∫ |𝑓(𝑥)| 𝑑𝑥 = ∫ 𝜆(𝑀𝑡 (𝑓)) 𝑑𝑡 , 𝑎

0

where 𝑀𝑐 (𝑓) is defined as in Exercise 0.1. More generally, prove that 𝑏

∞ 𝑝

∫ |𝑓(𝑥)| 𝑑𝑥 = 𝑝 ∫ 𝑡𝑝−1 𝜆(𝑀𝑡 (𝑓)) 𝑑𝑡 𝑎

0

for 𝑓 ∈ L𝑝 ([𝑎, 𝑏]), 𝑝 ≥ 1. Exercise 0.14. Let 𝑓 : [𝑎, 𝑏] → ℝ be monotonically increasing. Use Exercise 0.13 to prove that ∞

𝑓(𝑏)

∫ 𝐼𝑓 (𝑦) 𝑑𝑦 = ∫ 𝐼𝑓 (𝑦) 𝑑𝑦 = 𝑓(𝑏) − 𝑓(𝑎) , −∞

𝑓(𝑎)

where 𝐼𝑓 denotes the Banach indicatrix of 𝑓 introduced in Definition 0.38. Exercise 0.15. Let 𝑀 ⊂ ℝ be a measurable set and 𝑓 : 𝑀 → ℝ a measurable function. With the notation of Exercise 0.1, let ]|𝑓|[:= inf {𝑐 + 𝜆(𝑀𝑐 (𝑓)) : 𝑐 > 0} . Prove that ]|𝑓|[ is always finite if 𝑀 has finite measure, and construct an example which shows that ]|𝑓|[ may be infinite if 𝑀 has infinite measure.

0.5 Exercises to Chapter 0

| 49

Exercise 0.16. Prove that the quantity ]| ⋅ |[ defined in Exercise 0.15 satisfies the trian­ gle inequality ]|𝑓 + 𝑔|[≤]|𝑓|[+]|𝑔|[ . Moreover, use Luzin’s theorem (Theorem 0.2) to show that ]|𝑓|[= 0 if and only if 𝑓(𝑥) = 0 a.e. on 𝑀. Exercise 0.17. Show by means of an example that the quantity ]| ⋅ |[ defined in Exer­ cise 0.15 is not absolutely homogeneous, i.e. it is not true that ]|𝜆𝑓|[= |𝜆| ]|𝑓|[ for all 𝜆 ∈ ℝ, and so ]| ⋅ |[ is not a norm. Exercise 0.18. Given a measurable set 𝑀 ⊂ ℝ of finite measure, denote by 𝑆(𝑀) the linear space of all measurable functions 𝑓 : 𝑀 → ℝ, where we identify functions which coincide a.e. on 𝑀. Prove that 𝑑(𝑓, 𝑔) :=]|𝑓 − 𝑔|[, with ]| ⋅ |[ as in Exercise 0.15, defines a metric on 𝑆(𝑀). Moreover, prove that the metric space (𝑆(𝑀), 𝑑) is complete. Exercise 0.19. With the notation of Exercise 0.18, prove that convergence of a se­ quence (𝑓𝑛 )𝑛 in the metric 𝑑 coincides with convergence in measure (Exercise 0.11). Exercise 0.20. Find estimates for the constants 𝑚 and 𝑀 from (0.82) in the Orlicz space 𝑋 = L𝜙 introduced in Definition 0.18. Exercise 0.21. Let 𝜙 be some Young function and 𝜙∗ its conjugate Young function (0.23). On L𝜙 ([𝑎, 𝑏]), consider the Orlicz–Amemiya norm 𝑏

{1 } ⦀𝑓⦀L𝜙 := inf { ∫ 𝜙(𝜇𝑓(𝑥)) 𝑑𝑥 : 𝜇 > 0} . 𝜇 { 𝑎 } Show that this norm is equivalent to the Luxemburg norm (0.20) on L𝜙 ([𝑎, 𝑏]). Cal­ culate this norm in case 𝜙(𝑡) = 𝑡𝑝 , i.e. L𝜙 = L𝑝 . Exercise 0.22. Prove that the dual space to (𝐸𝜙 ([𝑎, 𝑏]), ‖⋅‖L 𝜙 ) is the space (L𝜙∗ ([𝑎, 𝑏]), ⦀⋅ ⦀L𝜙∗ ), where 𝐸𝜙 is the small Orlicz space defined in Section 0.1, 𝜙∗ denotes the conju­ gate Young function (0.23), and ⦀ ⋅ ⦀L𝜙 is the norm introduced in Exercise 0.21. More specifically, show that the map 𝛷(𝑔) := ℓ𝑔 with ℓ𝑔 given by (0.29), defines a linear surjective isometry between L𝜙∗ and 𝐸∗𝜙 . Exercise 0.23. Given 𝑓 ∈ L𝜙 ([𝑎, 𝑏]), denote by 𝑓𝑛 the truncation at height 𝑛 of 𝑓 de­ fined by {𝑓(𝑥) if |𝑓(𝑥)| ≤ 𝑛 , 𝑓𝑛 (𝑥) := { 0 if |𝑓(𝑥)| > 𝑛 . { Prove that lim ‖𝑓 − 𝑓𝑛 ‖L𝜙 = dist(𝑓, 𝐸𝜙 ) = inf {‖𝑓 − 𝑔‖ : 𝑔 ∈ 𝐸𝜙 } ,

𝑛→∞

50 | 0 Prerequisites where 𝐸𝜙 ([𝑎, 𝑏]) is the small Orlicz space defined in Section 0.1. Deduce that 𝐸𝜙 ([𝑎, 𝑏]) may be characterized as the closure of all essentially bounded functions on [𝑎, 𝑏] in the norm (0.11). Exercise 0.24. Prove that (L𝜙 ([𝑎, 𝑏]), ‖ ⋅ ‖L𝜙 ) is separable if and only if 𝜙 ∈ 𝛥 2 . Exercise 0.25. For 𝑓 : [𝑎, 𝑏] → ℝ, consider the following two statements: (a) there exists a bounded function 𝑔 : [𝑎, 𝑏] → ℝ such that 𝑓 ∼ 𝑔; (b) 𝑓 ∈ L∞ ([𝑎, 𝑏]), i.e. 𝑓 is essentially bounded on [𝑎, 𝑏]. Do any of these statements imply the other one? Exercise 0.26. For 𝑓 : [𝑎, 𝑏] → ℝ, consider the following two statements: (a) there exists a continuous function 𝑔 : [𝑎, 𝑏] → ℝ such that 𝑓 ∼ 𝑔; (b) 𝑓 is a.e. continuous on [𝑎, 𝑏]. Do any of these statements imply the other one? Exercise 0.27. Prove that the linear space of all polynomials 𝑝(𝑥) = 𝑎𝑛 𝑥𝑛 + 𝑎𝑛−1 𝑥𝑛−1 + . . . + 𝑎2 𝑥2 + 𝑎1 𝑥 + 𝑎0 with either the norm ‖𝑝‖ := |𝑎0 | + |𝑎1 | + . . . + |𝑎𝑛 | or the norm ⦀𝑝⦀ := max {|𝑎0 |, |𝑎1 |, . . . , |𝑎𝑛 |} is not a Banach space. Are these two norms equivalent? Is one of them equivalent to the norm (0.105)? Exercise 0.28. Suppose that (𝑋, ‖ ⋅ ‖𝑋 ) satisfies the hypotheses of Proposition 0.31, and let 𝑐(𝑋, 𝐵) be the sharp imbedding constant of 𝑋 󳨅→ 𝐵([𝑎, 𝑏]). Prove that (𝑋, ⦀ ⋅ ⦀𝑋 ) equipped with the norm ⦀𝑓⦀𝑋 := 2𝑐(𝑋, 𝐵)‖𝑓‖𝑋

(𝑓 ∈ 𝑋)

is then a Banach algebra. Exercise 0.29. Apply the preceding Exercise 0.28 to 𝑋 = Lip𝑜𝛼 ([𝑎, 𝑏]) and compare with Example 0.43. Exercise 0.30. Let (𝑋, ‖⋅‖𝑋 ) be a Banach algebra satisfying (0.37). Show that the norm ⦀ ⋅ ⦀𝑋 defined by³¹ ⦀𝑓⦀𝑋 := sup {‖𝑓𝑔‖𝑋 : ‖𝑔‖𝑋 = 1} is equivalent to ‖ ⋅ ‖𝑋 and satisfies (0.42).

31 Clearly, this norm is nothing else but the usual operator norm of the linear multiplication operator 𝑔 󳨃→ 𝑓𝑔 which, by (0.37), is bounded on 𝑋.

0.5 Exercises to Chapter 0 |

51

Exercise 0.31. Show that the supremum in (0.30) is attained in the following sense: for every function 𝑔 ∈ L𝑝󸀠 ([𝑎, 𝑏]) \ {0}, there exists a function 𝑓0 ∈ L𝑝 ([𝑎, 𝑏]) such that ‖𝑓0 ‖L𝑝 = 1 and 𝑏

‖𝑔‖L𝑝󸀠 = ∫ 𝑓0 (𝑥)𝑔(𝑥) 𝑑𝑥 . 𝑎

Moreover, prove that the function 𝑓0 is unique (up to equivalence). Exercise 0.32. Prove the formula (0.31) which is the analogue to (0.30) for 𝑝 = 1 and 𝑝󸀠 = ∞. Exercise 0.33. Show that the supremum in formula (0.31) is attained in the following sense: for every function 𝑔 ∈ L∞ ([𝑎, 𝑏]) \ {0}, there exists a function 𝑓0 ∈ L1 ([𝑎, 𝑏]) such that ‖𝑓0 ‖L1 = 1 and 𝑏

‖𝑔‖L∞ = ∫ 𝑓0 (𝑥)𝑔(𝑥) 𝑑𝑥 . 𝑎

Exercise 0.34. Show that the function 𝑓0 ∈ L1 ([𝑎, 𝑏]) from Exercise 0.33 is, in general, not unique. More precisely, find functions 𝑔 ∈ L∞ ([0, 1]) \ {0}, 𝑓1 ∈ L1 ([0, 1]), and 𝑓2 ∈ L1 ([0, 1]) such that ‖𝑓1 ‖L1 = ‖𝑓2 ‖L1 = 1 and 𝑓1 ≠ 𝑓2 , but 1

1

‖𝑔‖L∞ = ∫ 𝑓1 (𝑥)𝑔(𝑥) 𝑑𝑥 = ∫ 𝑓2 (𝑥)𝑔(𝑥) 𝑑𝑥 . 0

0

Exercise 0.35. Although Theorem 0.23 is not true for 𝑝 = ∞, prove that the formula (0.32) holds true which is the analogue to (0.30) for 𝑝 = ∞ and 𝑝󸀠 = 1. Exercise 0.36. Show that the supremum in formula (0.32) is attained in the following sense: for every function 𝑔 ∈ L1 ([𝑎, 𝑏]) \ {0}, there exists a function 𝑓0 ∈ L∞ ([𝑎, 𝑏]) such that ‖𝑓0 ‖L∞ = 1 and 𝑏

‖𝑔‖L1 = ∫ 𝑓0 (𝑥)𝑔(𝑥) 𝑑𝑥 . 𝑎

Exercise 0.37. Show that the function 𝑓0 ∈ L∞ ([𝑎, 𝑏]) from Exercise 0.36 is, in general, not unique. More precisely, find functions 𝑔 ∈ L1 ([0, 1]) \ {0}, 𝑓1 ∈ L∞ ([0, 1]), and 𝑓2 ∈ L∞ ([0, 1]) such that ‖𝑓1 ‖L∞ = ‖𝑓2 ‖L∞ = 1 and 𝑓1 ≠ 𝑓2 , but 1

1

‖𝑔‖L1 = ∫ 𝑓1 (𝑥)𝑔(𝑥) 𝑑𝑥 = ∫ 𝑓2 (𝑥)𝑔(𝑥) 𝑑𝑥 . 0

0

Exercise 0.38. Show that the equality holds in the Hölder inequality (0.14) for 1 < 󸀠 𝑝 < ∞ if and only if 𝛼|𝑓(𝑥)|𝑝 = 𝛽|𝑔(𝑥)|𝑝 (𝑎 ≤ 𝑥 ≤ 𝑏) for some constants 𝛼, 𝛽 ≥ 0 with 𝛼 + 𝛽 > 0.

52 | 0 Prerequisites Exercise 0.39*. Prove the following converse of Hölder’s inequality. Let 𝑔 : [𝑎, 𝑏] → ℝ be a measurable function, and suppose that 𝑓𝑔 ∈ L1 ([𝑎, 𝑏]) for every function 𝑓 ∈ L𝑝 ([𝑎, 𝑏]); then, 𝑔 ∈ L𝑝󸀠 ([𝑎, 𝑏]). Exercise 0.40. Show that the functional norm (0.25) may be equivalently defined by taking the supremum over ‖𝑥‖ < 1. Exercise 0.41. Denote by ℓ̂ : L∞ ([0, 1]) → ℝ the Hahn–Banach extension of the bounded linear functional ℓ : 𝐶([0, 1]) → ℝ in Example 0.24. Prove that there is no 𝑔 ∈ L1 ([0, 1]) such that ℓ̂ may be represented in the form (0.29). Exercise 0.42. Show that the dual space to (ℝ𝑛 , ‖ ⋅ ‖1 ), where ‖𝑥‖1 := |𝑥1 | + . . . + |𝑥𝑛 |, is isometrically isomorphic to the space (ℝ𝑛 , ‖ ⋅ ‖∞ ), where ‖𝑥‖∞ := max {|𝑥1 |, . . . , |𝑥𝑛 |}. Exercise 0.43. With the notation of Exercise 0.42, show that the dual space to (ℝ𝑛 , ‖ ⋅ ‖∞ ) is isometrically isomorphic to the space (ℝ𝑛 , ‖ ⋅ ‖1 ). Exercise 0.44. Show that the set 𝑆([𝑎, 𝑏]) of all step functions is a subalgebra of the normalized algebra (𝐵([𝑎, 𝑏]), ‖ ⋅ ‖∞ ). Exercise 0.45. Considering the function 𝑓𝛽 (𝑥) := |𝑥|𝛽 , show that the inclusion (0.72) is strict in case 𝛼 > 𝛽. More precisely, prove that for each 𝛽 ∈ (0, 1), there exists a function 𝑓 ∈ Lip𝛽 ([0, 1]) \ (⋃ Lip𝛼 ([0, 1])) . 𝛼>𝛽

Exercise 0.46. In analogy to Definition 0.44, define spaces Lip𝑛𝛼 ([𝑎, 𝑏]) for 0 < 𝛼 ≤ 1 and 𝑛 ∈ ℕ, and prove possible inclusions between them for varying 𝑛 and 𝛼. Exercise 0.47. Prove that the norms (0.63) and (0.64) are equivalent on 𝐶𝑛 ([𝑎, 𝑏]). Exercise 0.48. Prove that the norms (0.45) and (0.70) are not equivalent on Lip([𝑎, 𝑏]). Exercise 0.49. Prove that the norms (0.45) and (0.65) are not equivalent on 𝐶1 ([𝑎, 𝑏]). Exercise 0.50. Is the space Lip𝛼 ([𝑎, 𝑏]) with one of the norms (0.71) or (0.77) a Banach algebra? Exercise 0.51. Calculate the McShane extension (0.76) of the function 𝑓 ∈ Lip𝛼 ([0, 1]) defined by 𝑓(𝑥) := 𝑥𝛼 . Exercise 0.52. Given 𝛾 ∈ (0, 1), show that the function (0.86) belongs to Lip𝛾 ([0, 1]) if and only if 𝛼 is arbitrary and 𝛽 ≥ 1 − 𝛼/𝛾. Compare this with Proposition 0.48 (b). Exercise 0.53. For 𝑘 ∈ ℕ, calculate the norms (0.11), (0.45), (0.63), (0.64), (0.70), (0.71), (0.77), (0.78), and (0.79) of the function 𝑓𝑘 (𝑥) := 𝑥𝑘 over [0, 1].

0.5 Exercises to Chapter 0

| 53

Exercise 0.54. For 𝛼, 𝛽 ∈ ℝ with 𝛽 < 0, let 𝑓𝛼,𝛽 : [0, 1] → ℝ be defined by (0.86). Prove the following statements. (𝑛) (a) The 𝑛-th derivative 𝑓𝛼,𝛽 exists on [0, 1] if and only if 𝛼 > 1 + (𝑛 − 1)(1 − 𝛽). (𝑛) (b) The 𝑛-th derivative 𝑓𝛼,𝛽 exists and is bounded on [0, 1] if and only if 𝛼 ≥ 𝑛(1 − 𝛽).

(𝑛) (c) The 𝑛-th derivative 𝑓𝛼,𝛽 exists and is continuous on [0, 1] if and only if 𝛼 > 𝑛(1− 𝛽) .

Exercise 0.55. Let 𝑓𝛼,𝛽 : [0, 1] → ℝ be defined by (0.86). Find all pairs (𝛼, 𝛽) for which 𝑓𝛼,𝛽 belongs to Lip𝑛 ([0, 1]). Exercise 0.56. Show that the inclusion (0.72) is a continuous imbedding, i.e. Lip𝛼 ([𝑎, 𝑏]) 󳨅→ Lip𝛽 ([𝑎, 𝑏])

(0 < 𝛽 ≤ 𝛼 ≤ 1) ,

with sharp imbedding constant 𝑐(Lip𝛼 , Lip𝛽 ) = max {(𝑏 − 𝑎)𝛼−𝛽 , 1}. Exercise 0.57. Show that the two norms (0.78) and (0.79) are equivalent on Lip𝑛 ([𝑎, 𝑏]). Is Lip𝑛 ([𝑎, 𝑏]) with one of these norms a Banach algebra? Exercise 0.58. Calculate the constants 𝑚 and 𝑀 from (0.82) in the space 𝑋 = 𝐶𝑛 . Exercise 0.59. Calculate the constants 𝑚 and 𝑀 from (0.82) in the space 𝑋 = Lip𝑛 . Exercise 0.60. Calculate the constants 𝑚 and 𝑀 from (0.82) in the space 𝑋 = Lip𝑛𝛼 , see Exercise 0.46. Exercise 0.61. Calculate the constants 𝑚 and 𝑀 from (0.82) in the space 𝑋 = L𝑝 for 1 < 𝑝 < ∞. Exercise 0.62. Construct a function space which is not shift-invariant in the sense of Definition 0.45. Exercise 0.63. Show that the sharp imbedding constant for Lip([𝑎, 𝑏]) 󳨅→ Lip𝛼 ([𝑎, 𝑏]) is 𝑐(Lip, Lip𝛼 ) = max {(𝑏 − 𝑎)1−𝛼 , 1}. Compare with Exercise 0.56. Exercise 0.64. Show that the sharp imbedding constant for Lip𝛼 ([𝑎, 𝑏]) 󳨅→ 𝐶([𝑎, 𝑏]) is 𝑐(Lip𝛼 , 𝐶) = max {(𝑏 − 𝑎)𝛼 , 1}. Exercise 0.65. Let 𝑀 ⊂ ℝ be closed and 𝑓 : 𝑀 → [1, 2] continuous on 𝑀. Define 𝑓 ̂ : ℝ → ℝ by {𝑓(𝑥) ̂ := 𝑓(𝑥) { inf {𝑓(𝑦)|𝑥−𝑦|:𝑦∈𝑀} { inf {|𝑥−𝑦|:𝑦∈𝑀}

for 𝑥 ∈ 𝑀 , for 𝑥 ∈ ̸ 𝑀 .

̂ Show that 𝑓 ̂ is a continuous extension of 𝑓 with 𝑓(ℝ) ⊆ [1, 2]. Exercise 0.66. Prove that the sequence (𝑓𝑛 )𝑛 defined by 𝑓𝑛 (𝑥) := 12 sin 𝑛𝑥 converges in every Hölder space Lip𝛼 ([0, 1]) (0 < 𝛼 < 1) to zero, but not in the space Lip([0, 1]).

54 | 0 Prerequisites 1/4

Exercise 0.67. Prove that the sequence (𝑓𝑛 )𝑛 of 𝐶∞ -functions 𝑓𝑛 (𝑥) := (𝑥2 + 𝑛1 ) con­ verges in the space 𝐶([0, 1]) with norm (0.45) to 𝑓(𝑥) = √𝑥. Does it also converge in the Hölder space Lip1/2 ([0, 1]) with norm (0.71)? Exercise 0.68. Calculate the norms ‖𝑍𝐶,𝐷 ‖L1 and ‖𝑍𝜃 ‖L1 of the general zigzag function (0.91) and the special zigzag function (0.93) in L1 ([0, 1]). Exercise 0.69. If 𝑓 ∈ 𝐵([𝑎, 𝑏]) only has countably many points of discontinuity, does it follow that 𝑓 ∈ 𝑅([𝑎, 𝑏])? Exercise 0.70. Let 𝑓, 𝑔 ∈ 𝑆([𝑎, 𝑏]), and let (𝑓 ∨ 𝑔)(𝑥) := max {𝑓(𝑥), 𝑔(𝑥)} and (𝑓 ∧ 𝑔)(𝑥) := min {𝑓(𝑥), 𝑔(𝑥)}. Does it follow that 𝑓 ∨ 𝑔 ∈ 𝑆([𝑎, 𝑏]) and 𝑓 ∧ 𝑔 ∈ 𝑆([𝑎, 𝑏])? Exercise 0.71. Let 𝑓, 𝑔 ∈ 𝑅([𝑎, 𝑏]), and let 𝑓∨𝑔 and 𝑓∧𝑔 be defined as in Exercise 0.69. Does it follow that 𝑓 ∨ 𝑔 ∈ 𝑅([𝑎, 𝑏]) and 𝑓 ∧ 𝑔 ∈ 𝑅([𝑎, 𝑏])? Exercise 0.72. Show that the continuous imbeddings Lip𝛼 ([𝑎, 𝑏]) 󳨅→ Lip𝛼,𝑝 ([𝑎, 𝑏]) 󳨅→ Lip𝛼−1/𝑝 ([𝑎, 𝑏]) hold for 𝑝𝛼 > 1, where Lip𝛼,𝑝 denotes the generalized Hölder space introduced in Definition 0.54. Compare with Proposition 0.58. Exercise 0.73. In the terminology of Exercise 0.72, calculate the sharp imbedding constants 𝑐(Lip𝛼 , Lip𝛼,𝑝 ) and 𝑐(Lip𝛼,𝑝 , Lip𝛼−1/𝑝 ) for 𝑝𝛼 > 1. Exercise 0.74. Calculate the sharp imbedding constant 𝑐(Lip𝛼,𝑝 , Lip𝛽,𝑞 ), where 𝛼, 𝛽, 𝑝 and 𝑞 are as in Proposition 0.58. Exercise 0.75. Prove the following generalization of Theorem 0.42: given a set 𝑀 ⊂ ℝ and a modulus of continuity 𝜔 : [0, ∞) → [0, ∞), suppose that 𝑓 ∈ Lip𝜔,∞ (𝑀), see Definition 0.54. Then there exists a function 𝑓 ̂ ∈ Lip𝜔,∞ (ℝ) such that 𝑓|̂ 𝑀 = 𝑓 and lip𝜔,∞ (𝑓)̂ = lip𝜔,∞ (𝑓). Exercise 0.76. Suppose that a function 𝑓 ∈ L𝑝 ([𝑎, 𝑏]) satisfies 𝜔𝑝 (𝑓, [𝑎, 𝑏]; 𝛿) = 𝑜(𝛿)

(𝛿 → 0+) ,

where 𝜔𝑝 (𝑓, [𝑎, 𝑏]; 𝛿) denotes the integral modulus of continuity (0.98). Show that 𝑓 is constant a.e. on [𝑎, 𝑏].

1 Classical BV-spaces In this chapter, we start our analysis of the classical function space 𝐵𝑉([𝑎, 𝑏]) and its generalization 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) (1 ≤ 𝑝 < ∞). We discuss the basic properties of these spaces and their connections with other function spaces, with a particular empha­ sis on spaces of continuous, Lipschitz continuous, Hölder continuous, and absolutely continuous functions. Moreover, we will study the Jordan and Federer decomposi­ tions of 𝐵𝑉-functions, as well as Helly’s selection principle for bounded sequences of 𝐵𝑉-functions which, in a certain sense, is an analogue to the Arzelà–Ascoli compact­ ness criterion for sequences of continuous functions. In the last section, we briefly study functions of two variables and introduce the corresponding space 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) and related spaces.

1.1 Functions of bounded variation We start with the definition and properties of the classical function space 𝐵𝑉([𝑎, 𝑏]) which, as far as we know, goes back to Camille Jordan [153, 154]. Throughout the fol­ lowing, we denote by P([𝑎, 𝑏]) the family of all partitions of the interval [𝑎, 𝑏], i.e. all finite sets 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } (𝑚 ∈ ℕ variable) with 𝑎 = 𝑡0 < 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑚−1 < 𝑡𝑚 = 𝑏 .

(1.1)

𝜇(𝑃) := max {𝑡𝑗 − 𝑡𝑗−1 : 𝑗 = 1, 2, . . . , 𝑚}

(1.2)

The number is called the mesh size of the partition 𝑃. If 𝑡𝑗 − 𝑡𝑗−1 is independent of 𝑗, i.e. 𝑡1 − 𝑡0 = 𝑡2 − 𝑡1 = ⋅ ⋅ ⋅ = 𝑡𝑚 − 𝑡𝑚−1 =

𝑏−𝑎 , 𝑚

the partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } is called equidistant. Definition 1.1. Given a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) and a function 𝑓 : [𝑎, 𝑏] → ℝ, the nonnegative real number 𝑚

Var(𝑓, 𝑃) = Var(𝑓, 𝑃; [𝑎, 𝑏]) := ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|

(1.3)

𝑗=1

is called the variation (or Jordan variation) of 𝑓 on [𝑎, 𝑏] with respect to 𝑃. Moreover, the (possibly infinite) number Var(𝑓) = Var(𝑓; [𝑎, 𝑏]) := sup {Var(𝑓, 𝑃; [𝑎, 𝑏]) : 𝑃 ∈ P([𝑎, 𝑏])} ,

(1.4)

where the supremum is taken over all partitions of [𝑎, 𝑏], is called the total (Jordan) variation of 𝑓 on [𝑎, 𝑏]. In case Var(𝑓; [𝑎, 𝑏]) < ∞, we say that 𝑓 is a function of bounded variation (or function of bounded Jordan variation on [𝑎, 𝑏] and write 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]). ◼

56 | 1 Classical BV-spaces In this and subsequent chapters, we will need two important subsets of 𝐵𝑉([𝑎, 𝑏]) which we introduce right now. Definition 1.2. We denote by 𝐵𝑉𝑜 ([𝑎, 𝑏]) the subset of all functions 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) satisfying 𝑓(𝑎) = 0. Furthermore, we write 𝑁𝐵𝑉([𝑎, 𝑏]) for the set of all functions 𝑓 ∈ 𝐵𝑉𝑜 ([𝑎, 𝑏]) which are right continuous at any point 𝑥0 ∈ [𝑎, 𝑏), i.e. satisfy 𝑓(𝑥0 +) = lim 𝑓(𝑥) = 𝑓(𝑥0 ) 𝑥→𝑥0 +

(𝑎 ≤ 𝑥0 < 𝑏).

Functions in 𝑁𝐵𝑉([𝑎, 𝑏]) will be called normalized in what follows.

(1.5) ◼

For further use, we collect in the following proposition some important properties of the quantities (1.3) and (1.4). Proposition 1.3. The quantities (1.3) and (1.4) have the following properties. (a) The variation (1.4) is subadditive with respect to functions, i.e. Var(𝑓 + 𝑔; [𝑎, 𝑏]) ≤ Var(𝑓; [𝑎, 𝑏]) + Var(𝑔; [𝑎, 𝑏])

(1.6)

for 𝑓, 𝑔 : [𝑎, 𝑏] → ℝ. (b) The variation (1.4) is homogeneous with respect to functions, i.e. Var(𝜇𝑓; [𝑎, 𝑏]) = |𝜇| Var(𝑓; [𝑎, 𝑏])

(1.7)

|𝑓(𝑥) − 𝑓(𝑦)| ≤ Var(𝑓; [𝑥, 𝑦])

(1.8)

for 𝜇 ∈ ℝ. (c) The estimate holds for 𝑎 ≤ 𝑥 < 𝑦 ≤ 𝑏. (d) Every function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) is bounded with ‖𝑓‖∞ ≤ |𝑓(𝑎)| + Var(𝑓; [𝑎, 𝑏]),

(1.9)

where the norm ‖ ⋅ ‖∞ is given by (0.39). (e) Every monotone function 𝑓 : [𝑎, 𝑏] → ℝ belongs to 𝐵𝑉([𝑎, 𝑏]) with Var(𝑓; [𝑎, 𝑏]) = |𝑓(𝑏) − 𝑓(𝑎)| . (f) The variation (1.3) is monotone with respect to partitions, i.e. Var(𝑓, 𝑃; [𝑎, 𝑏]) ≤ Var(𝑓, 𝑄; [𝑎, 𝑏]) for 𝑃, 𝑄 ∈ P([𝑎, 𝑏]) with 𝑃 ⊆ 𝑄. (g) The variation (1.4) is additive with respect to intervals, i.e. Var(𝑓; [𝑎, 𝑏]) = Var(𝑓; [𝑎, 𝑐]) + Var(𝑓; [𝑐, 𝑏]) for 𝑎 < 𝑐 < 𝑏.

(1.10)

1.1 Functions of bounded variation

|

57

Proof. For two functions 𝑓, 𝑔 : [𝑎, 𝑏] → ℝ, 𝜇 ∈ ℝ, and any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), we have 𝑚

Var(𝑓 + 𝑔, 𝑃) = ∑ |(𝑓 + 𝑔)(𝑡𝑗 ) − (𝑓 + 𝑔)(𝑡𝑗−1 )| 𝑗=1 𝑚

𝑚

≤ ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| + ∑ |𝑔(𝑡𝑗 ) − 𝑔(𝑡𝑗−1 )| = Var(𝑓, 𝑃) + Var(𝑔, 𝑃) 𝑗=1

𝑗=1

and 𝑚

𝑚

Var(𝜇𝑓, 𝑃) = ∑ |(𝜇𝑓)(𝑡𝑗 ) − (𝜇𝑓)(𝑡𝑗−1 )| = |𝜇| ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| = |𝜇| Var(𝑓, 𝑃) , 𝑗=1

𝑗=1

which proves (a) and (b). To prove (c), it suffices to consider the special partition 𝑃 = {𝑥, 𝑦} ∈ P([𝑥, 𝑦]) and to observe that then |𝑓(𝑦) − 𝑓(𝑥)| = Var(𝑓, 𝑃; [𝑥, 𝑦]). Similarly, considering the partition 𝑃𝑥 = {𝑎, 𝑥, 𝑏} ∈ P([𝑎, 𝑏]), we obtain |𝑓(𝑥) − 𝑓(𝑎)| ≤ |𝑓(𝑏) − 𝑓(𝑥)| + |𝑓(𝑥) − 𝑓(𝑎)| = Var(𝑓, 𝑃𝑥 ; [𝑎, 𝑏]) , and hence |𝑓(𝑥)| ≤ |𝑓(𝑎)| + Var(𝑓, 𝑃𝑥 ; [𝑎, 𝑏]) ≤ |𝑓(𝑎)| + Var(𝑓; [𝑎, 𝑏]),

(1.11)

which proves (d) after taking the supremum over 𝑥 ∈ [𝑎, 𝑏]. Now, let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) and 𝑓 : [𝑎, 𝑏] → ℝ be increasing.¹ Then 𝑚

𝑚

Var(𝑓, 𝑃; [𝑎, 𝑏]) = ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| = ∑ [𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )] = 𝑓(𝑏) − 𝑓(𝑎) , 𝑗=1

𝑗=1

and hence Var(𝑓; [𝑎, 𝑏]) = 𝑓(𝑏) − 𝑓(𝑎) as well. In case of a decreasing function 𝑓 : [𝑎, 𝑏] → ℝ, the same argument shows that Var(𝑓; [𝑎, 𝑏]) = 𝑓(𝑎) − 𝑓(𝑏), and so we have proved (e). Clearly, the property (f) and the additivity formula in (g) follow directly from (1.3) and (1.4). It is clear that one cannot expect equality in (1.6); for example, for 𝑓(𝑡) = 𝑡 and 𝑔(𝑡) = −𝑡, we have Var(𝑓; [0, 1]) = Var(𝑔; [0, 1]) = 1, but Var(𝑓 + 𝑔; [0, 1]) = 0. Part (e) of Proposition 1.3 provides a link between monotone functions and func­ tions of bounded variation. One could ask if this could be in some sense inverted. The next example shows, however, that the converse of (e) is very far from being true: there exist functions of bounded variation which are not monotone on any interval.

1 We call a function 𝑓 increasing if 𝑥 < 𝑦 implies 𝑓(𝑥) ≤ 𝑓(𝑦), and decreasing if 𝑥 < 𝑦 implies 𝑓(𝑥) ≥ 𝑓(𝑦). So, in contrast to some authors, we do not require the strict inequalities 𝑓(𝑥) < 𝑓(𝑦) and 𝑓(𝑥) > 𝑓(𝑦), respectively. In that case, we call 𝑓 strictly increasing and strictly decreasing, respectively.

58 | 1 Classical BV-spaces Example 1.4. We arrange the rational numbers between 0 and 1 in a sequence, i.e. (0, 1) ∩ ℚ = {𝑟1 , 𝑟2 , 𝑟3 , . . . }, and define a function 𝑓 : [0, 1] → ℝ by {2−𝑘 𝑓(𝑡) := { 0 {

for 𝑡 = 𝑟𝑘 , otherwise .

Clearly, 𝑓 is not monotone on any interval [𝑎, 𝑏] ⊆ [0, 1]. On the other hand, we claim that Var(𝑓; [0, 1]) = 2, and hence 𝑓 ∈ 𝐵𝑉([0, 1]). To see this, we introduce some notation. For a bounded function 𝑓 : [𝑎, 𝑏] → ℝ and any subset 𝐴 ⊆ [𝑎, 𝑏], we call (1.12)

osc(𝑓; 𝐴) := sup 𝑓(𝑡) − inf 𝑓(𝑡) 𝑡∈𝐴

𝑡∈𝐴

the oscillation of 𝑓 on 𝐴. Let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } be an arbitrary partition of [0, 1]. For a fixed 𝑗 ∈ {1, 2, . . . , 𝑚}, denote 𝑘𝑗 := min{𝑖 : 𝑟𝑖 ∈ [𝑡𝑗−1 , 𝑡𝑗 ]} . Observe that every number in the finite sequence (𝑘1 , 𝑘2 , . . . , 𝑘𝑚 ) may appear at most two times, namely, as the right or left endpoint of two adjacent intervals gener­ ated by 𝑃. This yields 𝑚

𝑚

Var(𝑓, 𝑃; [0, 1]) = ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| ≤ ∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) 𝑗=1

𝑗=1

𝑚

∞

𝑗=1

𝑗=1

= ∑ 2−𝑘𝑗 ≤ 2 ∑ 2−𝑗 = 2, and so Var(𝑓; [0, 1]) ≤ 2, and hence 𝑓 ∈ 𝐵𝑉([0, 1]). To prove the equality Var(𝑓; [0, 1]) = 2, we construct a special partition in the following way. For fixed 𝑛 ∈ ℕ, we rearrange the first 𝑛 rational points 𝑟1 , 𝑟2 , . . . , 𝑟𝑛 in increasing order, and hence 𝑟1 < 𝑟2 < ⋅ ⋅ ⋅ < 𝑟𝑛 . Afterwards, we put 𝑠0 := 0,

𝑠1 := 𝑟1 ,

𝑠3 := 𝑟2 ,

𝑠5 := 𝑟3 ,

...

, 𝑠2𝑛−1 := 𝑟𝑛 ,

𝑠2𝑛 := 1

and choose irrational points 𝑠2 ∈ (𝑠1 , 𝑠3 ), 𝑠4 ∈ (𝑠3 , 𝑠5 ), 𝑠6 ∈ (𝑠5 , 𝑠7 ), . . . , 𝑠2𝑛−2 ∈ (𝑠2𝑛−3 , 𝑠2𝑛−1 ). For the corresponding partition 𝑃̃ = {𝑠0 , 𝑠1 , . . . , 𝑠2𝑛 } ∈ P([0, 1]), we get then 2𝑛

𝑗=1

2𝑛

2𝑛

1 − ( 12 )

𝑗=1

1−

Var(𝑓, 𝑃;̃ [0, 1]) = ∑ |𝑓(𝑠𝑗 ) − 𝑓(𝑠𝑗−1 )| = 2 ∑ 2−𝑗 =

1 2

= 2 (1 − 2−2𝑛 ) ,

which may be taken arbitrarily close to 2 by choosing 𝑛 sufficiently large. We conclude that the total variation of 𝑓 on [0, 1] is precisely 2. ♥

1.1 Functions of bounded variation

| 59

Now comes an important point. Although functions of bounded variation have no monotonicity behavior, there is a natural interconnection between bounded variation and monotonicity which is the statement of the classical Jordan decomposition theo­ rem [153]. Theorem 1.5. A function 𝑓 : [𝑎, 𝑏] → ℝ is of bounded variation if and only if it may be represented in the form 𝑓 = 𝑝𝑓 − 𝑛𝑓 , where both 𝑝𝑓 and 𝑛𝑓 are monotonically increasing functions. Proof. The fact that the sum or difference of two monotone functions has bounded variation is an immediate consequence of Proposition 1.3 (a) and (e); the nontrivial part is the converse. Given 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), consider the variation function 𝑉𝑓 : [𝑎, 𝑏] → ℝ defined by 𝑉𝑓 (𝑥) := Var(𝑓; [𝑎, 𝑥]) .

(1.13)

We take 𝑝𝑓 := 𝑉𝑓 . Clearly, 𝑝𝑓 is increasing with 𝑉𝑓 (𝑎) = 0 and 𝑉𝑓 (𝑏) = Var(𝑓; [𝑎, 𝑏]). So, the only point to show is that the function 𝑛𝑓 := 𝑉𝑓 − 𝑓 is increasing as well. However, for 𝑎 ≤ 𝑥 < 𝑦 ≤ 𝑏, we have 𝑓(𝑦) − 𝑓(𝑥) ≤ Var(𝑓; [𝑥, 𝑦]) = 𝑉𝑓 (𝑦) − 𝑉𝑓 (𝑥) , by (1.8) and (1.10), and so 𝑛𝑓 (𝑦) − 𝑛𝑓 (𝑥) = 𝑉𝑓 (𝑦) − 𝑓(𝑦) − 𝑉𝑓 (𝑥) + 𝑓(𝑥) ≥ 0 as claimed. In what follows, we will refer to the representation 𝑓 = 𝑉𝑓 − (𝑉𝑓 − 𝑓) constructed in Theorem 1.5 as the Jordan decomposition of 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]). The following result is a slight modification of Theorem 1.5. Theorem 1.6. A function 𝑓 : [𝑎, 𝑏] → ℝ is of bounded variation if and only if it may be represented in the form 𝑓 = 𝑝𝑓 − 𝑛𝑓 + 𝑓(𝑎), where both 𝑝𝑓 and 𝑛𝑓 are monotonically increasing nonnegative functions. Proof. We define 𝑝𝑓 , 𝑛𝑓 : [𝑎, 𝑏] → ℝ by 𝑝𝑓 (𝑥) :=

1 (𝑉 (𝑥) + 𝑓(𝑥) − 𝑓(𝑎)) , 2 𝑓

𝑛𝑓 (𝑥) :=

1 (𝑉 (𝑥) − 𝑓(𝑥) + 𝑓(𝑎)) 2 𝑓

with 𝑉𝑓 given by (1.13). By construction, we then have 𝑓(𝑥) = 𝑝𝑓 (𝑥) − 𝑛𝑓 (𝑥) + 𝑓(𝑎),

𝑉𝑓 (𝑥) = 𝑝𝑓 (𝑥) + 𝑛𝑓 (𝑥) .

Moreover, for 𝑎 ≤ 𝑥 < 𝑦 ≤ 𝑏, we get 𝑝𝑓 (𝑦) − 𝑝𝑓 (𝑥) =

1 (𝑉 (𝑦) + 𝑓(𝑦) − 𝑉𝑓 (𝑥) − 𝑓(𝑥)) ≥ 0 2 𝑓

60 | 1 Classical BV-spaces and

1 (𝑉 (𝑦) − 𝑓(𝑦) − 𝑉𝑓 (𝑥) + 𝑓(𝑥)) ≥ 0 , 2 𝑓 by Proposition 1.3 (e), which shows that 𝑝𝑓 and 𝑛𝑓 are in fact increasing. Since both 𝑝𝑓 (𝑎) = 0 and 𝑛𝑓 (𝑎) = 0, it follows that 𝑝𝑓 and 𝑛𝑓 are also nonnegative. 𝑛𝑓 (𝑦) − 𝑛𝑓 (𝑥) =

The variation function (1.13) employed in Theorems 1.5 and 1.6 has many interesting properties on its own. Moreover, some properties of 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) are “reflected” in 𝑉𝑓 , and vice versa (see Proposition 1.7 and Theorem 1.26 below). It may also be used to prove the following majorant principle: a function 𝑓 : [𝑎, 𝑏] → ℝ belongs to 𝐵𝑉([𝑎, 𝑏]) if and only if there exists an increasing function 𝑔 : [𝑎, 𝑏] → ℝ such that |𝑓(𝑥)−𝑓(𝑦)| ≤ 𝑔(𝑦) − 𝑔(𝑥) for any interval [𝑥, 𝑦] ⊆ [𝑎, 𝑏]. In fact, if such a function 𝑔 exists, we have Var(𝑓, 𝑃; [𝑎, 𝑏]) ≤ 𝑔(𝑏)−𝑔(𝑎) for any 𝑃 ∈ P([𝑎, 𝑏]), and hence 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]). Conversely, in case 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), we may simply choose 𝑔 := 𝑉𝑓 , as (1.8) shows. An important example of the “interaction” between 𝑓 and 𝑉𝑓 is given in the fol­ lowing Proposition 1.7. Suppose that 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) is continuous at some point 𝑥0 ∈ [𝑎, 𝑏]; then, the function 𝑉𝑓 from (1.13) is also continuous at 𝑥0 . The converse statement is also true. Proof. Let 𝜀 > 0, and suppose first that 𝑓 is continuous at 𝑥0 , where it is no loss of generality to assume that 𝑥0 < 𝑥 < 𝑏. Consider the difference 𝑉𝑓 (𝑥) − 𝑉𝑓 (𝑥0 ). Choose a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑥0 , 𝑏]) such that Var(𝑓; [𝑥0 , 𝑏]) < Var(𝑓, 𝑃; [𝑥0 , 𝑏]) + 𝜀 . Afterwards, we choose 𝛿 ∈ (0, 𝑡1 − 𝑥0 ) such that |𝑓(𝑥) − 𝑓(𝑥0 )| < 𝜀 for 0 < 𝑥 − 𝑥0 < 𝛿 which is possible by the continuity of 𝑓 at 𝑥0 . Then for these 𝑥, we have 𝑉𝑓 (𝑥) − 𝑉𝑓 (𝑥0 ) = Var(𝑓; [𝑥0 , 𝑥]) = Var(𝑓; [𝑥0 , 𝑏]) − Var(𝑓; [𝑥, 𝑏]) < Var(𝑓, 𝑃; [𝑥0 , 𝑏]) + 𝜀 − Var(𝑓; [𝑥, 𝑏]) 𝑚

≤ |𝑓(𝑥) − 𝑓(𝑥0 )| + |𝑓(𝑥) − 𝑓(𝑡1 )| + ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| − Var(𝑓; [𝑥, 𝑏]) + 𝜀 𝑗=2

≤ |𝑓(𝑥) − 𝑓(𝑥0 )| + 𝜀 < 2𝜀. This shows that 𝑉𝑓 (𝑥0 +) = 𝑉𝑓 (𝑥0 ), and so 𝑉𝑓 is right continuous at 𝑥0 . The left conti­ nuity of 𝑉𝑓 at 𝑥0 may be proved in the same way. To prove the converse statement, suppose now that 𝑉𝑓 is continuous at 𝑥0 ∈ [𝑎, 𝑏]. For 𝑥 ≥ 𝑥0 , we then have |𝑓(𝑥) − 𝑓(𝑥0 )| ≤ Var(𝑓; [𝑥0 , 𝑥]) = 𝑉𝑓 (𝑥) − 𝑉𝑓 (𝑥0 ) → 0

(𝑥 → 𝑥0 +) ,

while for 𝑥 ≤ 𝑥0 , we have |𝑓(𝑥) − 𝑓(𝑥0 )| ≤ Var(𝑓; [𝑥, 𝑥0 ]) = 𝑉𝑓 (𝑥0 ) − 𝑉𝑓 (𝑥) → 0 This proves the second assertion.

(𝑥 → 𝑥0 −) .

1.1 Functions of bounded variation

| 61

An alternative proof of Proposition 1.7 is contained in Exercise 1.6. In Theorem 1.26 below, we give a more systematic discussion of the “interplay” between the variation function 𝑉𝑓 and its parent function 𝑓. Theorems 1.5 and 1.6 explain why many “nice” properties of monotone functions (like Riemann integrability or differentiability a.e., see Exercises 1.28 and 1.29 carry over to functions of bounded variation. In particular, a function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) has, at most, many countably points of discontinuity in [𝑎, 𝑏], all being of first kind (jumps) or removable. On the other hand, the following well-known example shows that a continuous function need not have bounded variation. Example 1.8. Let 𝑓 : [0, 1] → ℝ be defined by {𝑥 sin 𝑥1 for 0 < 𝑥 ≤ 1 , (1.14) 𝑓(𝑥) := { 0 for 𝑥 = 0 . { Clearly, 𝑓 is continuous on [0, 1]. To show that 𝑓 ∈ ̸ 𝐵𝑉([0, 1]), we construct parti­ tions with the property that the oscillation sums in (1.3) become as large as possible. This may be achieved by choosing 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } in such a way that 𝑡𝑗−1 and 𝑡𝑗 give alternating maxima and minima of 𝑓, i.e. 2 2 , 𝑡2 := , ... , (2𝑚 − 1)𝜋 (2𝑚 − 3)𝜋 2 2 , 𝑡𝑚−1 := , 𝑡𝑚 := 1 . ... , 𝑡𝑚−2 := 5𝜋 3𝜋 In fact, we then have |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| ≥ 2𝑡𝑗−1 for 𝑗 = 2, 3, . . . , 𝑚 − 1, and so the sum in (1.3) becomes arbitrarily large if we choose 𝑚 sufficiently large.² ♥ 𝑡0 := 0 ,

𝑡1 :=

A refinement of Example 1.8 will be discussed in Exercises 1.8 and 1.9. In the next section, we will discuss relations between bounded variation and continuity in more detail. Now, we provide an important statement concerning the possibility of passing to the pointwise limit for a sequence of 𝐵𝑉-functions. Proposition 1.9. Let (𝑓𝑛 )𝑛 be a sequence in 𝐵𝑉([𝑎, 𝑏]) which converges pointwise on [𝑎, 𝑏] to some function 𝑓. Then Var(𝑓; [𝑎, 𝑏]) ≤ lim inf Var(𝑓𝑛 ; [𝑎, 𝑏]) . 𝑛→∞

(1.15)

Consequently, the pointwise limit of a sequence of functions with equibounded vari­ ations on the interval [𝑎, 𝑏] is a function of bounded variation on [𝑎, 𝑏]. Proof. If the right-hand side of (1.15) is infinite, there is nothing to prove. Thus, sup­ pose that lim inf Var(𝑓𝑛 ; [𝑎, 𝑏]) ≤ 𝐿 𝑛→∞

2 The reason is, of course, the divergence of the harmonic series.

62 | 1 Classical BV-spaces for some constant 𝐿 > 0. Then, keeping in mind the properties of the limit inferior of a real sequence, we can choose a subsequence (𝑓𝑛𝑘 )𝑘 of the sequence (𝑓𝑛 )𝑛 such that Var(𝑓𝑛𝑘 ; [𝑎, 𝑏]) ≤ 𝐿 for 𝑘 ∈ ℕ. Fix an arbitrary partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]). Then we obtain 𝑚

Var(𝑓𝑛𝑘 , 𝑃; [𝑎, 𝑏]) = ∑ |𝑓𝑛𝑘 (𝑡𝑗 ) − 𝑓𝑛𝑘 (𝑡𝑗−1 )| ≤ 𝐿 , 𝑗=1

and so, passing to the limit as 𝑘 → ∞, 𝑚

Var(𝑓, 𝑃; [𝑎, 𝑏]) = ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| ≤ 𝐿 . 𝑗=1

Since 𝐿 is independent of the partition 𝑃, this implies that Var(𝑓; [𝑎, 𝑏]) ≤ 𝐿, and the assertion follows. The last statement is an immediate consequence. In what follows, we shall describe the structure of the set 𝐵𝑉([𝑎, 𝑏]). First of all, let us observe that, by Proposition 1.3 (a) and (b), 𝐵𝑉([𝑎, 𝑏]) forms a real vector space with respect to the standard operations on functions. For an arbitrary function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), let us put (1.16) ‖𝑓‖𝐵𝑉 := |𝑓(𝑎)| + Var(𝑓; [𝑎, 𝑏]) . It is easy to check that (1.16) defines a norm on the linear space 𝐵𝑉([𝑎, 𝑏]). The following useful statement is concerned with the completeness of this space. Proposition 1.10. The space 𝐵𝑉([𝑎, 𝑏]) equipped with the norm (1.16) is a Banach space which is continuously imbedded into the space 𝐵([𝑎, 𝑏]) of all bounded functions on [𝑎, 𝑏] with norm (0.39). Moreover, 𝐵𝑉([𝑎, 𝑏]) is an algebra with Var(𝑓𝑔) ≤ ‖𝑓‖∞ Var(𝑔) + ‖𝑔‖∞ Var(𝑓)

(1.17)

for all 𝑓, 𝑔 ∈ 𝐵𝑉([𝑎, 𝑏]). Even better, we have ‖𝑓𝑔‖𝐵𝑉 ≤ ‖𝑓‖𝐵𝑉 ‖𝑔‖𝐵𝑉

(𝑓, 𝑔 ∈ 𝐵𝑉([𝑎, 𝑏])),

(1.18)

which means that 𝐵𝑉([𝑎, 𝑏]) is a normalized algebra. Proof. Assume that (𝑓𝑛 )𝑛 is a Cauchy sequence in 𝐵𝑉([𝑎, 𝑏]) with respect to the norm (1.16). Fix 𝜀 > 0 and choose a natural number 𝑛0 such that for all 𝑚, 𝑛 ∈ ℕ with 𝑚, 𝑛 ≥ 𝑛0 , we have ‖𝑓𝑛 − 𝑓𝑚 ‖𝐵𝑉 = |𝑓𝑛 (𝑎) − 𝑓𝑚 (𝑎)| + Var(𝑓𝑛 − 𝑓𝑚 ; [𝑎, 𝑏]) ≤ 𝜀 .

(1.19)

This inequality implies, in particular, |𝑓𝑛 (𝑎) − 𝑓𝑚 (𝑎)| ≤ 𝜀

(1.20)

1.1 Functions of bounded variation

|

63

for 𝑚, 𝑛 ≥ 𝑛0 , and so the real sequence (𝑓𝑛 (𝑎))𝑛 converges to some real number which we denote by 𝑓(𝑎). Letting 𝑚 → ∞ in (1.20), we get |𝑓𝑛 (𝑎) − 𝑓(𝑎)| ≤ 𝜀

(1.21)

for 𝑛 ≥ 𝑛0 . Further, from (1.19), we also obtain Var(𝑓𝑛 − 𝑓𝑚 ; [𝑎, 𝑏]) ≤ 𝜀 (𝑚, 𝑛 ≥ 𝑛0 ).

(1.22)

Now, fix an arbitrary point 𝑥 ∈ (𝑎, 𝑏]. Then, in view of (1.22) and Proposition 1.3 (g), we deduce that Var(𝑓𝑛 − 𝑓𝑚 ; [𝑎, 𝑥]) ≤ 𝜀. So, taking the special partition 𝑃𝑥 := {𝑎, 𝑥} of the interval [𝑎, 𝑥], we have 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨𝑓𝑛 (𝑥) − 𝑓𝑚 (𝑥)󵄨󵄨󵄨 − 󵄨󵄨󵄨𝑓𝑛 (𝑎) − 𝑓𝑚 (𝑎)󵄨󵄨󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨[𝑓𝑛 (𝑥) − 𝑓𝑚 (𝑥)] − [𝑓𝑛 (𝑎) − 𝑓𝑚 (𝑎)]󵄨󵄨󵄨 = Var(𝑓𝑛 − 𝑓𝑚 , 𝑃𝑥 ; [𝑎, 𝑥]) ≤ 𝜀, which yields 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨𝑓𝑛 (𝑥) − 𝑓𝑚 (𝑥)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝑓𝑛 (𝑎) − 𝑓𝑚 (𝑎)󵄨󵄨󵄨 + 𝜀 ≤ 2𝜀 (𝑎 ≤ 𝑥 ≤ 𝑏).

(1.23)

However, this means that for any fixed 𝑥 ∈ [𝑎, 𝑏], the sequence (𝑓𝑛 (𝑥))𝑛 is a Cauchy sequence, and so it has a pointwise limit which we denote by 𝑓(𝑥). Next, by fixing in (1.23) the index 𝑛 and letting 𝑚 → ∞, we obtain |𝑓𝑛 (𝑥) − 𝑓(𝑥)| ≤ 2𝜀 for 𝑛 ≥ 𝑛0 and 𝑥 ∈ [𝑎, 𝑏]. This shows that the function sequence (𝑓𝑛 )𝑛 is even uni­ formly convergent on [𝑎, 𝑏] to the function 𝑓. It remains to show that (𝑓𝑛 )𝑛 converges to 𝑓 in variation, i.e. in the norm (1.16). To this end, we first notice that (𝑓𝑛 )𝑛 is bounded in 𝐵𝑉([𝑎, 𝑏]), i.e. there exists a constant 𝑀 > 0 such that ‖𝑓𝑛 ‖𝐵𝑉 ≤ 𝑀 (𝑛 = 1, 2, . . . ) . In particular, we have Var(𝑓𝑛 ; [𝑎, 𝑏]) ≤ 𝑀 for all 𝑛 ∈ ℕ. Combining this with the last statement in Proposition 1.9 ensures that the limit function 𝑓 is of bounded variation on the interval [𝑎, 𝑏]. Further, observe that keeping 𝑛 fixed in (1.22) and passing to the limit inferior as 𝑚 → ∞, by virtue of Proposition 1.9, we get Var(𝑓𝑛 − 𝑓; [𝑎, 𝑏]) ≤ lim inf Var(𝑓𝑛 − 𝑓𝑚 ; [𝑎, 𝑏]) ≤ 𝜀 𝑚→∞

for 𝑛 ≥ 𝑛0 . This proves the completeness of 𝐵𝑉([𝑎, 𝑏]) in the norm (1.16). The fact that 𝐵𝑉 󳨅→ 𝐵 (with sharp imbedding constant 𝑐(𝐵𝑉, 𝐵) = 1) is only a reformulation of (1.9). Now, we prove that 𝐵𝑉([𝑎, 𝑏]) is an algebra satisfying (1.17). Fix 𝑓, 𝑔 ∈ 𝐵𝑉([𝑎, 𝑏]), and let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } be an arbitrary partition of [𝑎, 𝑏]. By Proposition 1.3 (d), both

64 | 1 Classical BV-spaces functions 𝑓 and 𝑔 are then bounded, and 𝑚

Var(𝑓𝑔, 𝑃) = ∑ |𝑓(𝑡𝑗 )𝑔(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )𝑔(𝑡𝑗−1 )| 𝑗=1 𝑚

𝑚

≤ ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| |𝑔(𝑡𝑗 )| + ∑ |𝑔(𝑡𝑗 ) − 𝑔(𝑡𝑗−1 )| |𝑓(𝑡𝑗−1 )| 𝑗=1

𝑗=1

≤ Var(𝑓, 𝑃) ‖𝑔‖∞ + Var(𝑔, 𝑃) ‖𝑓‖∞ , so by passing to the supremum with respect to 𝑃 ∈ P([𝑎, 𝑏]), we obtain (1.17). Finally, let us prove (1.18). Since 𝑓 󳨃→ Var(𝑓; [𝑎, 𝑏]) is a norm on the set 𝐵𝑉𝑜 ([𝑎, 𝑏]) of all 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) satisfying 𝑓(𝑎) = 0, from Proposition 0.31, we know that 𝐵𝑉𝑜([𝑎, 𝑏]) with norm ⦀𝑓⦀𝐵𝑉 = Var(𝑓) + ‖𝑓‖∞

(1.24)

is a normalized Banach algebra. However, we can get rid of both the term ‖𝑓‖∞ in (1.24) and the condition 𝑓(𝑎) = 0 by using the decomposition of 𝑓 given in Theorem 1.6. In fact, considering for 𝑓, 𝑔 ∈ 𝐵𝑉([𝑎, 𝑏]) the corresponding functions 𝑝𝑓𝑔 and 𝑛𝑓𝑔 of the product, we obtain ‖𝑓𝑔‖𝐵𝑉 = 𝑉𝑓𝑔 (𝑏) + |𝑓(𝑎)𝑔(𝑎)| = 𝑝𝑓𝑔 (𝑏) + 𝑛𝑓𝑔 (𝑏) + |𝑓(𝑎)𝑔(𝑎)| . Thus, using the subadditivity (1.6) of the total variation, we get ‖𝑓𝑔‖𝐵𝑉 ≤ 𝑝𝑓 (𝑏)𝑝𝑔 (𝑏) + 𝑝𝑓 (𝑏)𝑛𝑔 (𝑏) + |𝑔(𝑎)|𝑝𝑓 (𝑏) + 𝑛𝑓 (𝑏)𝑝𝑔 (𝑏) + 𝑛𝑓 (𝑏)𝑛𝑔 (𝑏) + |𝑔(𝑎)|𝑛𝑓 (𝑏) + |𝑓(𝑎)|𝑝𝑔 (𝑏) + |𝑓(𝑎)|𝑛𝑔 (𝑏) + |𝑓(𝑎)𝑔(𝑎)| = (𝑝𝑓 (𝑏) + 𝑛𝑓 (𝑏) + |𝑓(𝑎)|)(𝑝𝑔 (𝑏) + 𝑛𝑔 (𝑏) + |𝑔(𝑎)|) = ‖𝑓‖𝐵𝑉 ‖𝑔‖𝐵𝑉 which proves (1.18). Proposition 1.9 shows that finite variation carries over from a sequence to its point­ wise limit if this limit exists. Now, we prove a somewhat surprising statement about bounded sequences in 𝐵𝑉([𝑎, 𝑏]) which asserts the existence of such pointwise limits and is known as Helly’s selection principle. Theorem 1.11 (Helly). Let (𝑓𝑛 )𝑛 be a bounded sequence in 𝐵𝑉([𝑎, 𝑏]) with respect to the norm (1.16). Then (𝑓𝑛 )𝑛 contains a subsequence which converges pointwise on [𝑎, 𝑏] to some 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]). Proof. We use Theorem 1.5 to reduce the problem to monotone functions in the fol­ lowing way. Putting 𝑔𝑛 (𝑥) := 𝑉𝑓𝑛 (𝑥) = Var(𝑓𝑛 ; [𝑎, 𝑥]),

ℎ𝑛 (𝑥) := 𝑔𝑛 (𝑥) − 𝑓𝑛 (𝑥) ,

1.1 Functions of bounded variation

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65

we know that 𝑔𝑛 and ℎ𝑛 are increasing and bounded, and 𝑓𝑛 = 𝑔𝑛 −ℎ𝑛 for all 𝑛. We claim that the sequences (𝑔𝑛 )𝑛 and (ℎ𝑛 )𝑛 contain subsequences which converge pointwise on [𝑎, 𝑏] to some increasing function 𝑔 and ℎ, respectively; then, the function 𝑓 := 𝑔 − ℎ has the required properties. So, let 𝑔𝑛 : [𝑎, 𝑏] → ℝ be monotonically increasing with |𝑔𝑛 (𝑥)| ≤ 𝑐 < ∞ for all 𝑛 ∈ ℕ and 𝑎 ≤ 𝑥 ≤ 𝑏. Let 𝐸 := {𝑟1 , 𝑟2 , 𝑟3 , . . . } be some countable dense subset of [𝑎, 𝑏], where 𝑟1 := 𝑎 and 𝑟2 := 𝑏. Since |𝑔𝑛 (𝑟1 )| ≤ 𝑐 for all 𝑛, by the classical Bolzano–Weier­ strass theorem, we find a subsequence (𝑔𝑛(1) )𝑛 of (𝑔𝑛 )𝑛 which converges at the point 𝑟1 . Similarly, since |𝑔𝑛(1) (𝑟2 )| ≤ 𝑐 for all 𝑛, we find a subsequence (𝑔𝑛(2) )𝑛 of (𝑔𝑛(1) )𝑛 which converges at both points 𝑟1 and 𝑟2 . Having constructed (𝑔𝑛(𝑘−1) )𝑛 in this way, we choose a subsequence (𝑔𝑛(𝑘) )𝑛 of (𝑔𝑛(𝑘−1) )𝑛 which converges at all points 𝑟1 , 𝑟2 , . . . , 𝑟𝑘 . Thus, the diagonal sequence (𝑔𝑛𝑘 )𝑘 defined by 𝑔𝑛𝑘 (𝑥) := 𝑔𝑘(𝑘) (𝑥) converges at every point 𝑟𝑗 ∈ 𝐸. Now, we define 𝑔 : [𝑎, 𝑏] → ℝ by {lim𝑘→∞ 𝑔𝑛𝑘 (𝑥) 𝑔(𝑥) := { sup lim𝑘→∞ 𝑔𝑛𝑘 (𝑟𝑗 ) { 𝑟𝑗 0, we may choose 𝑟𝑖 , 𝑟𝑗 ∈ 𝐸 such that (1.26) 𝑟𝑖 < 𝑥 < 𝑟𝑗 , 𝑔(𝑟𝑗 ) − 𝑔(𝑟𝑖 ) < 𝜀 since 𝑔 is continuous at 𝑥. Moreover, by the pointwise convergence of (𝑔𝑛𝑘 )𝑘 to 𝑔 on 𝐸, we find some 𝑘0 ∈ ℕ such that |𝑔𝑛𝑘 (𝑟𝑖 ) − 𝑔(𝑟𝑖 )| < 𝜀 ,

|𝑔𝑛𝑘 (𝑟𝑗 ) − 𝑔(𝑟𝑗 )| < 𝜀

(1.27)

for 𝑘 ≥ 𝑘0 . Combining (1.26) and (1.27) yields 𝑔(𝑥) − 𝜀 < 𝑔(𝑟𝑖 ) ≤ 𝑔𝑛𝑘 (𝑟𝑖 ) + 𝜀 ≤ 𝑔𝑛𝑘 (𝑥) + 𝜀

(1.28)

≤ 𝑔𝑛𝑘 (𝑟𝑗 ) + 𝜀 < 𝑔(𝑟𝑗 ) + 2𝜀 ≤ 𝑔(𝑥) + 2𝜀 since 𝑟𝑖 < 𝑥 < 𝑟𝑗 and all functions occurring in (1.28) are increasing. This shows that 𝑔𝑛𝑘 (𝑥) → 𝑔(𝑥) pointwise on [𝑎, 𝑏]\𝐷 as 𝑘 → ∞. However, the set 𝐷 is at most countable and the sequence (𝑔𝑛𝑘 )𝑘 is uniformly bounded. Thus, we may again use the diagonal procedure described above and find another subsequence which converges pointwise on the whole interval [𝑎, 𝑏]. Clearly, the limit function 𝑔 obtained in this way is increas­ ing (Exercise 1.17), and so we are done. Interestingly, Helly’s selection principle may also be used to give an alternative proof of the completeness of the space (𝐵𝑉([𝑎, 𝑏]), ‖ ⋅ ‖𝐵𝑉 ). To see this, let (𝑓𝑛 )𝑛 be a Cauchy

66 | 1 Classical BV-spaces sequence with respect to the norm (1.16); without loss of generality, we assume that 𝑓𝑛 (𝑎) = 0 since otherwise, we pass from 𝑓𝑛 to the function 𝑓𝑛 −𝑓𝑛 (𝑎). Given 𝜀 > 0, choose 𝑛0 ∈ ℕ such that (1.22) holds. By Helly’s selection principle (Theorem 1.11), we find a subsequence (𝑓𝑛𝑘 )𝑘 of (𝑓𝑛 )𝑛 which converges pointwise on [𝑎, 𝑏] to some 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]). For any partition 𝑃 ∈ P([𝑎, 𝑏]) and all 𝑚 ≥ 𝑛0 , we then get Var(𝑓𝑚 − 𝑓, 𝑃; [𝑎, 𝑏]) = lim Var(𝑓𝑚 − 𝑓𝑛𝑘 , 𝑃; [𝑎, 𝑏]) ≤ 𝜀 , 𝑘→∞

and hence ‖𝑓𝑚 − 𝑓‖𝐵𝑉 ≤ 𝜀 for 𝑚 ≥ 𝑛0 , which proves the assertion. Obviously, the monotone functions on a fixed interval [𝑎, 𝑏] do not form a linear space, as may easily be seen by considering 𝑓(𝑡) = 𝑡2 and 𝑔(𝑡) = 1 − 𝑡 on [𝑎, 𝑏] = [0, 1], say (see also Exercise 1.11). Theorem 1.5 and Proposition 1.3 (a), (b) and (e) show that 𝐵𝑉([𝑎, 𝑏]) is the linear hull (or span) of the monotone functions, i.e. the smallest lin­ ear space which contains all monotone functions. This is another way of introducing functions of bounded variation. We remark that the problem of whether or not the quotient 𝑓/𝑔 of two functions 𝑓, 𝑔 ∈ 𝐵𝑉([𝑎, 𝑏]) belongs again to 𝐵𝑉([𝑎, 𝑏]) is more delicate, see Exercises 1.1 and 1.2. Moreover, it is illuminating to compare the two conditions 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) and |𝑓| ∈ 𝐵𝑉([𝑎, 𝑏]), where |𝑓| is defined by |𝑓|(𝑥) := |𝑓(𝑥)|, see Exercises 1.3–1.5. In Chapter 5 of this monograph, we will study the so-called composition operator problem in great detail for the space 𝐵𝑉([𝑎, 𝑏]) (and many similar function spaces), which consists of determining conditions, possibly both necessary and sufficient, on a function 𝑔 : ℝ → ℝ, under which the composition 𝑔 ∘ 𝑓 belongs to 𝐵𝑉([𝑎, 𝑏]) when­ ever 𝑓 belongs to 𝐵𝑉([𝑎, 𝑏]). Here, we briefly discuss the dual substitution operator problem of characterizing all “admissible changes of variables,” i.e. the problem of determining conditions on a transformation 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑], under which 𝑓 ∘ 𝜏 be­ longs to 𝐵𝑉([𝑎, 𝑏]) for all 𝑓 ∈ 𝐵𝑉([𝑐, 𝑑]). For example, we have the following necessary and sufficient condition in case of an increasing bijection 𝜏. Proposition 1.12. Given a function 𝑔 : [𝑐, 𝑑] → ℝ, let 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] be continuous and strictly increasing³ with 𝜏(𝑎) = 𝑐 and 𝜏(𝑏) = 𝑑. Then 𝑔 ∘ 𝜏 ∈ 𝐵𝑉([𝑎, 𝑏]) if and only if 𝑔 ∈ 𝐵𝑉([𝑐, 𝑑]). Proof. Given a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), the set 𝜏(𝑃) = {𝜏(𝑡0 ), 𝜏(𝑡1 ), . . . , 𝜏(𝑡𝑚 )} is then a partition of [𝑐, 𝑑], by our assumptions on 𝜏. Therefore, in case 𝑔 ∈ 𝐵𝑉([𝑐, 𝑑]), we have Var(𝑔 ∘ 𝜏, 𝑃; [𝑎, 𝑏]) = Var(𝑔, 𝜏(𝑃); [𝑐, 𝑑]) ,

3 The classical intermediate value theorem implies that under these hypotheses, 𝜏 is then a homeo­ morphism, i.e. a bijection with continuous inverse.

1.1 Functions of bounded variation

| 67

and hence Var(𝑔 ∘ 𝜏; [𝑎, 𝑏]) ≤ Var(𝑔; [𝑐, 𝑑]) . Applying this reasoning to the function 𝜏−1 : [𝑐, 𝑑] → [𝑎, 𝑏] (which has the same properties as 𝜏), we conclude that also Var(𝑔; [𝑐, 𝑑]) = Var(𝑔 ∘ 𝜏 ∘ 𝜏−1 ; [𝑐, 𝑑]) ≤ Var(𝑔 ∘ 𝜏; [𝑎, 𝑏]) . This shows that Var(𝑔; [𝑐, 𝑑]) = Var(𝑔 ∘ 𝜏; [𝑎, 𝑏]), and so the assertion follows. Our proof shows that under the hypotheses of Proposition 1.12, the map 𝑔 󳨃→ 𝑔 ∘ 𝜏 is a (surjective) isometry between the two function spaces (𝐵𝑉([𝑎, 𝑏]), ‖ ⋅ ‖𝐵𝑉 ) and (𝐵𝑉([𝑐, 𝑑]), ‖ ⋅ ‖𝐵𝑉 ) since 𝑔(𝜏(𝑎)) = 𝑔(𝑐) and 𝑔(𝜏(𝑏)) = 𝑔(𝑑). Observe that the “if” part in the statement of Proposition 1.12 is also valid if 𝜏 is merely monotone, but not necessarily continuous. In fact, if 𝑔 : [𝑐, 𝑑] → ℝ has bounded variation, we may decompose 𝑔, by Theorem 1.5, as difference 𝑔 = 𝑝𝑔 − 𝑛𝑔 , where both 𝑝𝑔 and 𝑛𝑔 are increasing. If 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] is increasing (respectively decreasing), then 𝑔 ∘ 𝜏 = 𝑝𝑔 ∘ 𝜏 − 𝑛𝑔 ∘ 𝜏, where both 𝑝𝑔 ∘ 𝜏 and 𝑛𝑔 ∘ 𝜏 are increasing (re­ spectively decreasing), and so 𝑔 ∘ 𝜏 : [𝑎, 𝑏] → ℝ has bounded variation as well. On the other hand, the following simple example shows that one cannot drop the continuity assumption on 𝜏 in the “only if” part of Proposition 1.12. Example 1.13. Define 𝜏 : [0, 4] → [0, 4] by 𝜏(0) := 0 and 𝜏(𝑡) := 3 + 14 𝑡 for 0 < 𝑡 ≤ 4. Then 𝜏 is strictly increasing with 𝜏(0) = 0 and 𝜏(4) = 4, but discontinuous at 𝑡 = 0. The function 𝑔 : [0, 4] → ℝ defined by 0 { { { 𝑔(𝑥) := {tan 𝜋2 (𝑥 − 1) { { {0

for 0 ≤ 𝑥 ≤ 1 , for 1 < 𝑥 < 2 , for 2 ≤ 𝑥 ≤ 4

does not belong to 𝐵𝑉([0, 4]) since it is unbounded near 𝑥 = 2. On the other hand, the function (𝑔 ∘ 𝜏)(𝑡) ≡ 0 trivially belongs to 𝐵𝑉([0, 4]). ♥ Proposition 1.12 shows that, roughly speaking, homeomorphisms are suitable changes of variables which preserve bounded variation. Since being homeomorphic is a strong property, this is not surprising. So the question arises whether or not one can describe the precise class of “admissible changes of variables” which preserve bounded varia­ tion. This leads to a new class of maps which has been introduced, as far as we know, by Josephy [155]. Definition 1.14. For 𝑛 = 1, 2, 3, . . . , we denote by J𝑛 the family of sets 𝑀 ⊆ ℝ which may be represented as a union of 𝑛 intervals.⁴ Obviously, J𝑛 ⊂ J𝑛+1 since each interval is a union of two nonempty subintervals.

4 Here, the intervals may be closed, open, half-open, or singletons.

68 | 1 Classical BV-spaces By 𝐽𝑛 ([𝑎, 𝑏]), we denote the class of all bounded functions 𝑓 : [𝑎, 𝑏] → ℝ such that 𝑓−1 ([𝛼, 𝛽]) ∈ J𝑛 for any interval⁵ [𝛼, 𝛽] ⊂ ℝ. Moreover, we put 𝐽([𝑎, 𝑏]) := ⋃ 𝐽𝑛 ([𝑎, 𝑏])

(1.29)

𝑛

and call functions 𝑓 ∈ 𝐽([𝑎, 𝑏]) pseudomonotone in what follows.

◼

The following proposition shows that pseudomonotone functions are intermediate be­ tween monotone functions and functions of bounded variation. Proposition 1.15. Every monotone function is pseudomonotone, and every pseudomono­ tone function has bounded variation. Proof. The pseudomonotonicity of a monotone function 𝑓 follows from the fact that 𝑓−1 ([𝛼, 𝛽]) is always an interval (closed, open, or half-open) if 𝑓 is monotonically in­ creasing or decreasing,⁶ and so 𝑓−1 ([𝛼, 𝛽]) ∈ J1 for any interval [𝛼, 𝛽] ⊂ ℝ. Now, let 𝑓 ∈ 𝐽𝑛 ([𝑎, 𝑏]) for some 𝑛 ∈ ℕ; we show that Var(𝑓; [𝑎, 𝑏]) ≤ 4(𝑛 + 1)‖𝑓‖∞ ,

(1.30)

where ‖ ⋅ ‖∞ denotes the norm (0.39). If (1.30) is false, we find a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) such that 𝑚

∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| > 4(𝑛 + 1)‖𝑓‖∞ .

(1.31)

𝑗=1

Some nondegenerate interval [𝐴, 𝐵] ⊆ [−‖𝑓‖∞ , ‖𝑓‖∞ ] is covered more than 2(𝑛+1) times by subintervals [𝑓(𝑡𝑗−1 ), 𝑓(𝑡𝑗 )] or [𝑓(𝑡𝑗 ), 𝑓(𝑡𝑗−1 )] of our partition 𝑃 in (1.31), and at least 𝑛 + 1 among them are of type [𝑓(𝑡𝑗−1 ), 𝑓(𝑡𝑗 )]. Consequently, 𝑛+1

[𝐴, 𝐵] ⊆ ⋂ [𝑓(𝑥𝑘 ), 𝑓(𝑦𝑘 )] , 𝑘=1

where 𝑥𝑘 = 𝑡𝑗−1 and 𝑦𝑘 = 𝑡𝑗 for some 𝑗 ∈ {1, 2, . . . , 𝑚}, and 𝑥1 < 𝑦1 < ⋅ ⋅ ⋅ < 𝑥𝑛+1 < 𝑦𝑛+1 , and so 𝑓(𝑥𝑘 ) ≤ 𝐴 < 𝐵 ≤ 𝑓(𝑦𝑘 ) (𝑘 = 1, 2, . . . , 𝑛 + 1) . Now, choosing 𝛼 and 𝛽 such that 𝐴 < 𝛼 < 𝐵 and 𝛽 > max {𝑓(𝑦1 ), . . . , 𝑓(𝑦𝑛+1 )}, we see that 𝑓(𝑥𝑘 ) belongs to [𝛼, 𝛽] for all 𝑘, while 𝑓(𝑦𝑘 ) does not belong to [𝛼, 𝛽] for any 𝑘. This means that 𝑓−1 ([𝛼, 𝛽]) ∈ ̸ J𝑛 , contradicting our assumption. 5 Without loss of generality, we take into account only intervals [𝛼, 𝛽] satisfying 𝑓−1 ([𝛼, 𝛽])∩[𝑎, 𝑏] ≠ 0. 6 The fact that 𝑓−1 (𝐼) is an interval for each interval 𝐼 is even equivalent to the monotonicity of a function 𝑓, see Exercise 1.10. Of course, for strictly monotone functions, we simply have 𝑓−1 ([𝛼, 𝛽]) = [𝑓−1 (𝛼), 𝑓−1 (𝛽)] if 𝑓 is increasing, and 𝑓−1 ([𝛼, 𝛽]) = [𝑓−1 (𝛽), 𝑓−1 (𝛼)] if 𝑓 is decreasing. If 𝑓 is not strictly monotone, however, the interval 𝑓−1 ([𝛼, 𝛽]) need not be closed.

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| 69

Finding a pseudomonotone function which is not monotone is easy. In the following example,⁷ we give a function 𝑓 ∈ 𝐵𝑉([0, 1]) which is not pseudomonotone. Example 1.16. Let 𝑓 : [0, 1] → ℝ be defined by {𝑥2 sin2 𝑓(𝑥) := { 0 {

1 𝑥

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 .

(1.32)

A straightforward calculation shows⁸ that Var(𝑓; [0, 1]) ≤ 4, and hence 𝑓 ∈ 𝐵𝑉([0, 1]). On the other hand, 1 𝑓−1 ({0}) = {0} ∪ { 𝑛𝜋 : 𝑛 = 1, 2, 3, . . . } ,

♥

and so 𝑓 ∈ ̸ 𝐽([0, 1]).

Now, we come to the announced refinement of Proposition 1.12 which gives a precise description of all admissible changes of variables. Proposition 1.17. Let 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] be given. Then 𝑔 ∘ 𝜏 ∈ 𝐵𝑉([𝑎, 𝑏]) for all 𝑔 ∈ 𝐵𝑉([𝑐, 𝑑]) if and only if 𝜏 is pseudomonotone. Proof. Suppose first that 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] is pseudomonotone, and let 𝑔 : [𝑐, 𝑑] → ℝ be increasing. Similarly, as in the proof of Proposition 1.15, one may show then that 𝜏−1 ([𝛼, 𝛽]) ∈ 𝐽𝑛 implies (𝑔 ∘ 𝜏)−1 ([𝛼, 𝛽]) = 𝜏−1 (𝑔−1 ([𝛼, 𝛽])) ∈ 𝐽𝑛 , and so 𝑔 ∘ 𝜏 ∈ 𝐵𝑉([𝑎, 𝑏]). Now, a general function 𝑔 ∈ 𝐵𝑉([𝑐, 𝑑]) may be represented as the difference 𝑔 = 𝑝𝑔 −𝑛𝑔 of increasing functions 𝑝𝑓 and 𝑛𝑓 , by Theorem 1.5, and so we have 𝑔∘𝜏 = 𝑝𝑔 ∘𝜏−𝑛𝑔 ∘𝜏 ∈ 𝐵𝑉([𝑎, 𝑏]). Suppose now that 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] is not pseudomonotone, and hence 𝜏 ∈ ̸ 𝐽𝑛 ([𝑎, 𝑏]) for all 𝑛. We construct a function 𝑔 ∈ 𝐵𝑉([𝑐, 𝑑]) such that 𝑔 ∘ 𝜏 ∈ ̸ 𝐵𝑉([𝑎, 𝑏]). By assumption, we can find a sequence (𝐼𝑛 )𝑛 of intervals 𝐼𝑛 ⊆ [𝑐, 𝑑] such that 𝜏−1 (𝐼𝑛 ) ∈ ̸ J3𝑛

(𝑛 = 1, 2, 3, . . . ) .

(1.33)

Define 𝑔𝑛 : [𝑐, 𝑑] → ℝ and 𝑔 : [𝑐, 𝑑] → ℝ by 𝑔𝑛 (𝑥) :=

𝜒𝐼𝑛 (𝑥) 3𝑛

∞

,

𝑔(𝑥) := ∑ 𝑔𝑛 (𝑥) . 𝑛=1

𝑛

Then Var(𝑔𝑛 ; [𝑐, 𝑑]) ≤ 2/3 , and hence ∞

∞

1 = 1, 𝑛 3 𝑛=1

Var(𝑔; [𝑐, 𝑑]) ≤ ∑ Var(𝑔𝑛 ; [𝑐, 𝑑]) ≤ 2 ∑ 𝑛=1

7 The idea of Example 1.16 is plane: pseudomonotonicity of 𝑓 means that 𝑓, apart from constant pieces, does not repeat its values “too often.” 8 By considering the derivative of 𝑓, one may even show that 𝑓 is Lipschitz continuous on [0, 1], and so has bounded variation, see (1.46) in the next section.

70 | 1 Classical BV-spaces and so 𝑔 ∈ 𝐵𝑉([𝑐, 𝑑]). On the other hand, Var(𝑔𝑛 ∘ 𝜏; [𝑎, 𝑏]) = Var(3−𝑛 𝜒𝜏−1 (𝐼𝑛 ) ; [𝑎, 𝑏]) ≥ 3𝑛

2 = 2. 3𝑛

We claim that this implies Var(𝑔 ∘ 𝜏; [𝑎, 𝑏]) ≥

1 ∞ ∑ Var(𝑔𝑛 ∘ 𝜏; [𝑎, 𝑏]) = ∞ , 6 𝑛=1

(1.34)

and so 𝑔 ∘ 𝜏 ∈ ̸ 𝐵𝑉([𝑎, 𝑏]). In fact, suppose that 𝑔𝑛 ∘ 𝜏 is discontinuous at some point 𝑥0 ∈ [𝑎, 𝑏], but 𝑔1 ∘ 𝜏, 𝑔2 ∘ 𝜏, . . . 𝑔𝑛−1 ∘ 𝜏 are all continuous at 𝑥0 . Then 𝑥0 is a point of discontinuity for 𝑔 ∘ 𝜏, contributing a jump of at least ∞ 1 1 1 − ∑ 𝑘 = 𝑛 3 2 ⋅ 3𝑛 3 𝑘=𝑛+1

in 𝑔 ∘ 𝜏, but contributing no more than ∞

1 1 = 𝑛−1 𝑘 3 3 𝑘=𝑛

2∑

to the series in (1.34). From this, the assertion follows. We remark that other decomposition results with admissible changes of variables will be given in Theorem 1.28 and Theorem 1.41 below. We close this section with a useful result which shows that if a function 𝑓 : [𝑎, 𝑏] → ℝ fails to have bounded variation, it always fails “locally:” Proposition 1.18 (localization principle). Suppose that 𝑓 ∈ ̸ 𝐵𝑉([𝑎, 𝑏]). Then there ex­ ists a point 𝑥0 ∈ [𝑎, 𝑏] such that 𝑓 ∈ ̸ 𝐵𝑉([𝑐, 𝑑]) for each interval [𝑐, 𝑑] ⊆ [𝑎, 𝑏] such that 𝑐 < 𝑥0 < 𝑑. Proof. Suppose that the assertion is false, which means that for each 𝑥 ∈ [𝑎, 𝑏], there exists some open interval 𝐼𝑥 containing 𝑥 such that 𝑓 ∈ 𝐵𝑉(𝐼𝑥 ). Since [𝑎, 𝑏] ⊆ ⋃ 𝐼𝑥 𝑎≤𝑥≤𝑏

and [𝑎, 𝑏] is compact, we may find finitely many points 𝑥1 , . . . , 𝑥𝑛 ∈ [𝑎, 𝑏] such that 𝑛

[𝑎, 𝑏] ⊆ ⋃ 𝐼𝑥𝑘 . 𝑘=1

However, then, Proposition 1.3 (g) implies 𝑛

Var(𝑓; [𝑎, 𝑏]) ≤ ∑ Var(𝑓; 𝐼𝑥𝑘 ) < ∞ , 𝑘=1

contradicting our assumption.

1.2 Bounded variation and continuity

| 71

1.2 Bounded variation and continuity Let us now return to continuity properties of functions of bounded variation. As in Chapter 0, we write 𝐶([𝑎, 𝑏]) for the linear space of all continuous functions on [𝑎, 𝑏], equipped with the natural norm (0.45). As we have seen, neither of the spaces 𝐶([𝑎, 𝑏]) or 𝐵𝑉([𝑎, 𝑏]) is contained in the other. In some applications, however, the function space 𝐶([𝑎, 𝑏]) ∩ 𝐵𝑉([𝑎, 𝑏]) is of some interest. This space is related to another subclass of 𝐵𝑉([𝑎, 𝑏]) which we will discuss now. Given 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), consider the quantity Var0 (𝑓) defined by Var0 (𝑓) = Var0 (𝑓; [𝑎, 𝑏]) := lim sup {Var(𝑓) − Var(𝑓, 𝑃) : 𝜇(𝑃) ≤ 𝛿},

(1.35)

𝛿→0+

where 𝜇(𝑃) denotes the mesh size (1.2) of 𝑃. Roughly speaking, the quantity (1.35) mea­ sures the degree of uniform approximability of the total variation (1.4) by choosing partitions 𝑃 in (1.3) of sufficiently small mesh size. In order to represent Var0 (𝑓) in a more transparent form, note that the quantity 𝑊(𝛿) defined by 𝑊(𝛿) := Var(𝑓) − inf {Var(𝑓, 𝑃) : 𝜇(𝑃) ≤ 𝛿}

(𝛿 > 0)

is increasing on (0, ∞), and so has a limit for 𝛿 → 0+, which is nothing else but (1.35). In what follows, let us denote by 𝐶𝐵𝑉([𝑎, 𝑏]) the subset of all functions 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) such that Var0 (𝑓; [𝑎, 𝑏]) = 0. A straightforward calculation shows that 𝐶𝐵𝑉([𝑎, 𝑏]) contains all monotone functions on [𝑎, 𝑏]. Unfortunately, 𝐶𝐵𝑉([𝑎, 𝑏]) is not a linear subspace of 𝐵𝑉([𝑎, 𝑏]): Example 1.19. In the space 𝐵𝑉([0, 2]), for example, let 𝑓 = 𝜒[0,1] be the characteristic function of the interval [0, 1], and 𝑔 = 𝜒[1,2] be the characteristic function of the interval [1, 2]. Clearly, Var(𝑓) = Var(𝑔) = 1 and, for any partition 𝑃 of [0, 2], we have Var(𝑓, 𝑃) = Var(𝑔, 𝑃) = 1. This implies that Var0 (𝑓) = Var0 (𝑔) = 0, and so 𝑓, 𝑔 ∈ 𝐶𝐵𝑉([0, 2]). On the other hand, it is easily seen that 𝑓 + 𝑔 = 1 + 𝜒{1} , and therefore Var(𝑓 + 𝑔) = 2 and ♥ Var0 (𝑓 + 𝑔) = 2, so 𝑓 + 𝑔 ∈ ̸ 𝐶𝐵𝑉([0, 2]). A similar choice of 𝑓 and 𝑔 shows that the set 𝐶𝐵𝑉([𝑎, 𝑏]) is not convex, so it has a rather poor behavior from the algebraic viewpoint. Since 𝐶𝐵𝑉([𝑎, 𝑏]) contains all monotone functions, its convex hull (equivalently, linear hull) coincides with the whole space 𝐵𝑉([𝑎, 𝑏]), by Theorem 1.5 or 1.6. The following Proposition 1.20 provides a link between the class 𝐶𝐵𝑉([𝑎, 𝑏]) and the continuous functions of bounded variation. Proposition 1.20. Let 𝑓 be a continuous function of bounded variation on [𝑎, 𝑏]. Then 𝑓 belongs to 𝐶𝐵𝑉([𝑎, 𝑏]). Proof. Given 𝜀 > 0, by definition (1.4) of the total variation, we may find a partition 𝑃0 = {𝜏0 , 𝜏1 , . . . , 𝜏𝑛 } ∈ P([𝑎, 𝑏]) such that Var(𝑓, 𝑃0 ) ≥ Var(𝑓) − 𝜀.

(1.36)

72 | 1 Classical BV-spaces Since 𝑓 is uniformly continuous on [𝑎, 𝑏], we may further choose a 𝛿 > 0 such that for 𝑠, 𝑡 ∈ [𝑎, 𝑏] with |𝑠 − 𝑡| ≤ 𝛿, we have |𝑓(𝑠) − 𝑓(𝑡)| ≤

𝜀 . 2(𝑛 − 1)

(1.37)

Now, let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) be an arbitrary partition satisfying 𝜇(𝑃) ≤ 𝛿. Consider the partition 𝑃 ∪ {𝜏𝑖 } which differs from 𝑃 by adding just one point 𝜏𝑖 from 𝑃0 . If this point lies in the interval (𝑡𝑗−1 , 𝑡𝑗 ) for some 𝑗 ∈ {1, 2, . . . , 𝑚}, then the enlarged partition has the form 𝑃 ∪ {𝜏𝑖 } = {𝑡0 , 𝑡1 , . . . , 𝑡𝑗−1 , 𝜏𝑖 , 𝑡𝑗 , . . . , 𝑡𝑚 }, and so we have Var(𝑓, 𝑃 ∪ {𝜏𝑖 }) 𝑗−1

= ∑ |𝑓(𝑡𝑘 ) − 𝑓(𝑡𝑘−1 )| + |𝑓(𝜏𝑖 ) − 𝑓(𝑡𝑗−1 )| 𝑘=1 𝑚

+ |𝑓(𝑡𝑗 ) − 𝑓(𝜏𝑖 )| + ∑ |𝑓(𝑡𝑘 ) − 𝑓(𝑡𝑘−1 )| 𝑘=𝑗+1

= Var(𝑓, 𝑃) + |𝑓(𝜏𝑖 ) − 𝑓(𝑡𝑗−1 )| + |𝑓(𝑡𝑗 ) − 𝑓(𝜏𝑖 )| − |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| ≤ Var(𝑓, 𝑃) + |𝑓(𝜏𝑖 ) − 𝑓(𝑡𝑗−1 )| + |𝑓(𝑡𝑗 ) − 𝑓(𝜏𝑖 )| . Since both |𝜏𝑖 − 𝑡𝑗−1 | ≤ 𝛿 and |𝑡𝑗 − 𝜏𝑖 | ≤ 𝛿, in view of (1.37), we get Var(𝑓, 𝑃 ∪ {𝜏𝑖 }) ≤ Var(𝑓, 𝑃) +

𝜀 , 𝑛−1

showing that adding one new point 𝜏𝑖 ∈ 𝑃0 to 𝑃 leads to an increase of Var(𝑓, 𝑃) by at most 𝜀/(𝑛 − 1). Since there are not more than 𝑛 − 1 points in 𝑃0 which are different from the points of 𝑃, we conclude that Var(𝑓, 𝑃 ∪ 𝑃0 ) − Var(𝑓, 𝑃) ≤ (𝑛 − 1)

𝜀 = 𝜀. 𝑛−1

Combining this inequality with (1.36), we obtain Var(𝑓, 𝑃) ≥ Var(𝑓, 𝑃 ∪ 𝑃0 ) − 𝜀 ≥ Var(𝑓, 𝑃0 ) − 𝜀 ≥ Var(𝑓) − 2𝜀 . This means that Var0 (𝑓) ≤ 2𝜀, and so 𝑓 ∈ 𝐶𝐵𝑉([𝑎, 𝑏]) since 𝜀 > 0 is arbitrary. We may reformulate the statement of Proposition 1.20 as inclusion 𝐶([𝑎, 𝑏]) ∩ 𝐵𝑉([𝑎, 𝑏]) ⊆ 𝐶𝐵𝑉([𝑎, 𝑏]) ⊆ 𝐵𝑉([𝑎, 𝑏]) .

(1.38)

The function 𝑓 in Example 1.19 shows that the first inclusion in (1.38) is strict, while the function 𝑓 + 𝑔 in Example 1.19 shows that the second inclusion in (1.38) is strict. To the best of our knowledge, the first characterization of the condition Var0 (𝑓) = 0 was apparently given by Tonelli [304] (cf. also [54]) who showed that 𝑓 ∈ 𝐶𝐵𝑉([𝑎, 𝑏]) if and only if (𝑓(𝑡+) − 𝑓(𝑡))(𝑓(𝑡) − 𝑓(𝑡−)) ≥ 0 (𝑎 ≤ 𝑡 ≤ 𝑏) ,

1.2 Bounded variation and continuity

|

73

i.e. 𝑓 keeps “jumping in the same direction” at any point of discontinuity. From this result, it follows not only that every continuous function 𝑓 of bounded variation be­ longs to the class 𝐶𝐵𝑉([𝑎, 𝑏]), but also every monotone function on [𝑎, 𝑏], as already observed above. Moreover, this condition again explains the geometrical meaning of Example 1.19. We may formulate the assertion of Proposition 1.20 in the following in a slightly different way. Given a continuous function 𝑓 : [𝑎, 𝑏] → ℝ and a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), consider the oscillation of 𝑓 defined in (1.12) on each subinterval of 𝑃. Then the equality {𝑚 } Var(𝑓; [𝑎, 𝑏]) = lim { ∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) : 𝜇(𝑃) ≤ 𝛿} (1.39) 𝛿→0+ 𝑗=1 { } holds, where the variation of 𝑓 on [𝑎, 𝑏] may be, of course, infinite; see Exercise 1.39. We remark that other sufficient conditions for a function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) to belong to 𝐶𝐵𝑉([𝑎, 𝑏]) may be found in [13]. Let us now consider two important classes of functions on [𝑎, 𝑏] which are con­ tained in 𝐶([𝑎, 𝑏]) ∩ 𝐵𝑉([𝑎, 𝑏]), and so also in 𝐶𝐵𝑉([𝑎, 𝑏]), by (1.38). The first class will be studied in great detail in Chapter 3. We denote by 𝛴([𝑎, 𝑏]) the family of all finite collections 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛]} of pairwise nonoverlapping subintervals⁹ of [𝑎, 𝑏], and by 𝛴∞ ([𝑎, 𝑏]) the family of all countably infinite collections 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} of pairwise nonoverlapping subintervals of [𝑎, 𝑏]. Definition 1.21. A function 𝑓 : [𝑎, 𝑏] → ℝ is called absolutely continuous if it has the following property: for any 𝜀 > 0, there exists a 𝛿 > 0 such that for all collections 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏]), the condition 𝑛

∑ (𝑏𝑘 − 𝑎𝑘 ) ≤ 𝛿

(1.40)

𝑘=1

implies that

𝑛

∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| ≤ 𝜀.

(1.41)

𝑘=1

Equivalently, we may require that for any 𝜀 > 0, there exists a 𝛿 > 0 such that for all 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]) satisfying ∞

∑ (𝑏𝑘 − 𝑎𝑘 ) ≤ 𝛿,

(1.42)

𝑘=1

we have

∞

∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| ≤ 𝜀.

(1.43)

𝑘=1

We denote the set of all absolutely continuous functions on [𝑎, 𝑏] by 𝐴𝐶([𝑎, 𝑏]).

◼

9 Two intervals 𝐼 and 𝐽 are called nonoverlapping if 𝐼𝑜 ∩ 𝐽𝑜 = 0, i.e. they are disjoint or have at most one boundary point in common.

74 | 1 Classical BV-spaces Let us briefly explain the word “equivalently” in Definition 1.21. Of course, the con­ dition involving 𝛴∞ ([𝑎, 𝑏]) implies the condition involving 𝛴([𝑎, 𝑏]). Conversely, sup­ pose that the first requirement in Definition 1.21 is fulfilled, and suppose that 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]) satisfies (1.42). Then (1.40) holds true for any fixed 𝑛, and so also (1.41). However, this implies that (1.43) holds true as well. Of course, absolute continuity implies (uniform) continuity on [𝑎, 𝑏]. Moreover, it is not hard to see that every absolutely continuous function has bounded variation: Proposition 1.22. Every function 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) belongs to 𝐵𝑉([𝑎, 𝑏]). Proof. In fact, for 𝜀 = 1, say, choose 𝛿 > 0 such that (1.40) implies (1.41), and con­ sider the equidistant partition 𝑃𝑛 := {𝑡0 , 𝑡1 , . . . , 𝑡𝑛 }, where 𝑛 ∈ ℕ is so large that 𝑛𝛿 > 𝑏 − 𝑎. On each subinterval [𝑡𝑘−1 , 𝑡𝑘 ] of this partition, we then have, by construction, Var(𝑓, 𝑃𝑛 ; [𝑡𝑘−1 , 𝑡𝑘 ]) ≤ 1. By the additivity property of the variation (Proposition 1.3 (g)), we then get Var(𝑓, 𝑃𝑛 ; [𝑎, 𝑏]) ≤ 𝑛, and thus also Var(𝑓; [𝑎, 𝑏]) ≤ 𝑛. Another important class of continuous functions which is closely related to the space 𝐵𝑉([𝑎, 𝑏]) is the space of Lipschitz continuous functions introduced in Definition 0.39. Recall that the linear space 𝐿𝑖𝑝([𝑎, 𝑏]) with norm ‖𝑓‖𝐿𝑖𝑝 := |𝑓(𝑎)| + 𝑙𝑖𝑝(𝑓) , where 𝑙𝑖𝑝(𝑓) := sup 𝑥=𝑦 ̸

|𝑓(𝑥) − 𝑓(𝑦)| |𝑥 − 𝑦|

(1.44)

(1.45)

is a Banach space. Clearly, every Lipschitz continuous function is absolutely continu­ ous (choose 𝛿 := 𝜀/𝑙𝑖𝑝(𝑓)). Thus, we may extend the chain of inclusions (1.38) to 𝐿𝑖𝑝([𝑎, 𝑏]) ⊆ 𝐴𝐶([𝑎, 𝑏]) ⊆ 𝐶([𝑎, 𝑏]) ∩ 𝐵𝑉([𝑎, 𝑏]) ⊆ 𝐵𝑉([𝑎, 𝑏]).

(1.46)

One may even show that the space (𝐿𝑖𝑝([𝑎, 𝑏]), ‖ ⋅ ‖𝐿𝑖𝑝 ) is continuously imbedded into the space (𝐵𝑉([𝑎, 𝑏]), ‖ ⋅ ‖𝐵𝑉 ) in the sense of Definition 0.29. In fact, suppose that 𝑓 : [𝑎, 𝑏] → ℝ satisfies a Lipschitz condition |𝑓(𝑥) − 𝑓(𝑦)| ≤ 𝐿|𝑥 − 𝑦| (𝑎 ≤ 𝑥, 𝑦 ≤ 𝑏) for some 𝐿 > 0, and let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } be any partition of [𝑎, 𝑏]. Then 𝑚

𝑚

Var(𝑓, 𝑃; [𝑎, 𝑏]) = ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| ≤ 𝐿 ∑ |𝑡𝑗 − 𝑡𝑗−1 | = 𝐿(𝑏 − 𝑎) , 𝑗=1

𝑗=1

and hence ‖𝑓‖𝐵𝑉 = |𝑓(𝑎)| + Var(𝑓; [𝑎, 𝑏]) ≤ |𝑓(𝑎)| + 𝐿(𝑏 − 𝑎),

(1.47)

‖𝑓‖𝐵𝑉 ≤ max {1, 𝑏 − 𝑎} ‖𝑓‖𝐿𝑖𝑝

(1.48)

and so

1.2 Bounded variation and continuity

|

75

since 𝐿 may be taken arbitrarily close to 𝑙𝑖𝑝(𝑓). On the other hand, the following non­ trivial example shows that none of the Hölder spaces 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) introduced in Defini­ tion 0.39 is contained in 𝐵𝑉([𝑎, 𝑏]) in case 𝛼 < 1. Example 1.23. Fix 𝛼 ∈ (0, 1). We construct a function 𝑓 ∈ 𝐿𝑖𝑝𝛼 ([0, 1]) \ 𝐵𝑉([0, 1]). To this end, we define a constant 𝛾 and a sequence (𝑡𝑛 )𝑛 in [0, 1] by¹⁰ ∞

𝛾 := 𝜁(1/𝛼, 0) = ∑ 𝑘=1

1 𝑘1/𝛼

,

𝑡𝑛 :=

1 ∞ 1 ∑ 𝛾 𝑘=𝑛 𝑘1/𝛼

(1.49)

for 𝑛 = 1, 2, 3, . . . , and define 𝑓 : [0, 1] → ℝ by 0 for 𝑥 = 0 , { { { 𝑛 𝑓(𝑥) := { (−1) for 𝑥 = 𝑡𝑛 , { { 𝑛 {linear otherwise. Considering, similarly as in Example 1.8, partitions 𝑃𝑛 which contain the points 𝑡1 , 𝑡2 , . . . , 𝑡𝑛 , we see that 𝑛

1 → ∞ (𝑛 → ∞) , 𝑘 𝑘=1

Var(𝑓, 𝑃𝑛 ; [0, 1]) ≥ ∑

and so 𝑓 ∈ ̸ 𝐵𝑉([0, 1]). Now, let 0 < 𝑥 < 𝑦 ≤ 1, and choose 𝑚, 𝑛 ∈ ℕ such that 𝑡𝑛+1 ≤ 𝑥 ≤ 𝑡𝑛 and 𝑡𝑚+1 ≤ 𝑦 ≤ 𝑡𝑚 . We distinguish three cases. Suppose first that 𝑛 = 𝑚. Then 0 < 𝑦 − 𝑥 ≤ 𝑡𝑛 − 𝑡𝑛+1 =

1 , 𝛾𝑛1/𝛼

and hence |𝑓(𝑥) − 𝑓(𝑦)| = (𝑦 − 𝑥)

|𝑓(𝑡𝑛 ) − 𝑓(𝑡𝑛+1 )| 2𝑛1/𝛼 ≤ (𝑦 − 𝑥)𝛾 |𝑡𝑛 − 𝑡𝑛+1 | 𝑛 ≤ 2𝛾|𝑥 − 𝑦|𝛼 |𝑡𝑛 − 𝑡𝑛+1 |1−𝛼 𝑛(1−𝛼)/𝛼 1 ≤ 2𝛾|𝑥 − 𝑦|𝛼 1−𝛼 𝑛(1−𝛼)/𝛼−(1−𝛼)/𝛼 = 2𝛾𝛼 |𝑥 − 𝑦|𝛼 . 𝛾

Assume now that 𝑛 = 𝑚 + 1. Then 𝑡𝑛 = 𝑡𝑚+1 , and hence |𝑓(𝑥) − 𝑓(𝑦)| ≤ |𝑓(𝑥) − 𝑓(𝑡𝑛 )| + |𝑓(𝑡𝑚+1 ) − 𝑓(𝑦)| ≤ 2𝛾𝛼 (|𝑥 − 𝑡𝑛 |𝛼 + |𝑡𝑚+1 − 𝑦|𝛼 ) ≤ 4𝛾𝛼 |𝑥 − 𝑦|𝛼 .

10 In (1.49) and (1.51) below, we use the shortcut (0.17).

76 | 1 Classical BV-spaces Finally, let 𝑛 ≥ 𝑚 + 2. Then we find points 𝑠 ∈ [𝑡𝑛 , 𝑡𝑛−1 ] and 𝑡 ∈ [𝑡𝑚+2 , 𝑡𝑚+1 ] such that¹¹ 𝑓(𝑠) = 𝑓(𝑡) = 0. Consequently, |𝑓(𝑥) − 𝑓(𝑦)| ≤ |𝑓(𝑥) − 𝑓(𝑠)| + |𝑓(𝑠) − 𝑓(𝑡)| + |𝑓(𝑡) − 𝑓(𝑦)| ≤ 4𝛾𝛼 (|𝑥 − 𝑠|𝛼 + |𝑡 − 𝑦|𝛼 ) ≤ 8𝛾𝛼 |𝑥 − 𝑦|𝛼 . In each case, we see that 𝑓 satisfies the Hölder condition (0.67) with 𝐿 := 8𝛾𝛼 for 0 < 𝑥 < 𝑦 ≤ 1. Since 𝑓 is continuous at 0 with 𝑓(0) = 0, the same is true on the interval [0, 1]. ♥ The function in Example 1.23 looks very much like a “mirror reversed” copy of the zigzag function (0.91) introduced in Definition 0.49. In fact, we could have also used (0.91) as an example of a function in 𝐿𝑖𝑝𝛼 ([0, 1]) \ 𝐵𝑉([0, 1]). As we have seen in Propo­ sition 0.50, we have 𝑍𝐶,𝐷 ∈ 𝐿𝑖𝑝𝛼 ([0, 1]) if and only if 𝑆𝛼 (𝐶, 𝐷) = sup {𝑑𝑛 𝑐𝑛−𝛼 : 𝑛 = 1, 2, 3, . . . } < ∞ . However, in this case, we have 𝑑𝑛 ≤ 𝑆𝑐𝑛𝛼 for all 𝑛 ∈ ℕ. Therefore, from the obvious relation ∞

Var(𝑍𝐶,𝐷 ; [0, 1]) = ∑ 𝑑𝑘 ,

(1.50)

𝑘=1

it follows that for constructing such an example we should have ∞

∞

∑ 𝑐𝑘𝛼 = ∞,

∑ 𝑐𝑘 = 𝑆𝛼 (𝐶, 𝐷)1/𝛼 < ∞ .

𝑘=1

𝑘=1

The simplest choice of the 𝑐𝑘 ’s with this property is of course 𝑐𝑘 = 𝑘−𝛼 , and this explains Definition (1.49). The function 𝑓 in Example 1.23 belongs to 𝑓 ∈ 𝐿𝑖𝑝𝛼 ([0, 1]) \ 𝐵𝑉([0, 1]) with pre­ scribed 𝛼 < 1. The following is a refinement of Example 1.23. Example 1.24. In this example, we construct a function 𝑓 ∈ ( ⋂ 𝐿𝑖𝑝𝛼 ([0, 1])) \ 𝐵𝑉([0, 1]) . 0 0 be given, and choose 𝛿 > 0 such that (1.40) implies 𝑛 󵄨 󵄨 ∑ 󵄨󵄨󵄨󵄨𝑉𝑓 (𝑏𝑘 ) − 𝑉𝑓 (𝑎𝑘 )󵄨󵄨󵄨󵄨 ≤ 𝜀

𝑘=1

80 | 1 Classical BV-spaces for 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏]). Then 𝑛

𝑛

∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| ≤ ∑ (𝑉𝑓 (𝑏𝑘 ) − 𝑉𝑓 (𝑎𝑘 )) ≤ 𝜀 𝑘=1

𝑘=1

which shows that 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]). The proof is complete. Theorem 1.26 shows that the variation function 𝑉𝑓 inherits several properties from its parent function 𝑓. Observe, however, that there is a certain asymmetry in statement (d); in fact, the question as to if 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) implies 𝑉𝑓 ∈ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) for 0 < 𝛼 < 1 seems to be open.¹³ Of course, the mere requirement 𝑓 ∈ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) does not imply 𝑉𝑓 ∈ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) since 𝑉𝑓 need not be finite, as Example 1.23 shows. However, we do not know of any example where 𝑉𝑓 is finite for some 𝑓 ∈ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]), but 𝑉𝑓 ∈ ̸ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]). The reader may also have noticed that differentiability does not occur in Theo­ rem 1.26. The reason is simple: differentiability does not carry over in any direction, see Exercises 1.50 and 1.51. As was pointed out in [149], if 𝑓 has bounded variation on [𝑎, 𝑏], then both 𝑓 and 𝑉𝑓 are differentiable on [𝑎, 𝑏], except possibly on a nullset. However, the set of points at which 𝑓󸀠 (𝑥) exists is not necessarily the same as the set of points at which 𝑉𝑓󸀠 (𝑥) exists. Interestingly, in [281], it is shown that 𝑉𝑓󸀠 (𝑥) = |𝑓󸀠 (𝑥)| a.e. on [𝑎, 𝑏], see Exercise 1.52. Now, we establish a connection between the variation of a continuous function and the Lebesgue integral of its Banach indicatrix, see Definition 0.38 or (0.106). Proposition 1.27. Let 𝑓 : [𝑎, 𝑏] → ℝ be continuous, and let 𝐼𝑓 : ℝ → ℕ0 ∪ {∞} denote its Banach indicatrix defined by (0.106). Then the equality ∞

(1.53)

Var(𝑓; [𝑎, 𝑏]) = ∫ 𝐼𝑓 (𝑦) 𝑑𝑦 −∞

holds. In particular, 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) if and only if 𝐼𝑓 ∈ 𝐿 1 (ℝ). Proof. For 𝑛 = 1, 2, 3, . . . and 𝑘 = 1, 2, . . . , 2𝑛−1 , we put 𝛿𝑘,𝑛 := 𝑘(𝑏 − 𝑎)/2𝑛 and 𝛥 1,𝑛 := [𝑎, 𝑎 + 𝛿1,𝑛 ) , ... ,

𝛥 2,𝑛 := [𝑎 + 𝛿1,𝑛 , 𝑎 + 𝛿2,𝑛 ) ,

𝛥 2𝑛−1 ,𝑛 := [𝑎 + 𝛿2𝑛−1 −1,𝑛 , 𝑎 + 𝛿2𝑛−1 ,𝑛 ) ,

...

𝛥 2𝑛 ,𝑛 := [𝑎 + 𝛿2𝑛−1 ,𝑛 , 𝑏] .

Moreover, for 𝑘 = 1, 2, . . . , 2𝑛 , let us denote 𝑚𝑘,𝑛 := inf {𝑓(𝑥) : 𝑥 ∈ 𝛥 𝑘,𝑛 } ,

𝑀𝑘,𝑛 := sup {𝑓(𝑥) : 𝑥 ∈ 𝛥 𝑘,𝑛 } ,

and define functions 𝑔𝑘,𝑛 : ℝ → {0, 1} and 𝑔𝑛 : ℝ → ℕ0 by {1 𝑔𝑘,𝑛 (𝑦) := 𝜒𝑓−1 (𝑦)∩𝛥 𝑘,𝑛 (𝑦) = { 0 {

13 For a related question, see Exercise 1.25.

if 𝑓(𝑥) = 𝑦 for some 𝑥 ∈ 𝛥 𝑘,𝑛 , otherwise

(1.54)

1.2 Bounded variation and continuity

and

| 81

2𝑛

𝑔𝑛 (𝑦) := ∑ 𝑔𝑘,𝑛 (𝑦) . 𝑘=1

Clearly, 𝑔𝑛+1 (𝑦) ≥ 𝑔𝑛 (𝑦) for all 𝑦 ∈ ℝ. We claim that the sequence (𝑔𝑛 )𝑛 converges pointwise on ℝ to the Banach indicatrix 𝐼𝑓 of 𝑓. Indeed, suppose first that 𝐼𝑓 (𝑦) = 𝑚, and let 𝑓−1 (𝑦) = {𝑥1 , 𝑥2 , . . . , 𝑥𝑚 } denote the set of all solutions of the equation 𝑓(𝑥) = 𝑦 in [𝑎, 𝑏]. Choose 𝑁 ∈ ℕ so large that 2−𝑁 < min {𝑥𝑖 − 𝑥𝑗 : 1 ≤ 𝑖, 𝑗 ≤ 𝑚, 𝑖 ≠ 𝑗}; then, for 𝑛 > 𝑁, all elements in 𝑓−1 (𝑦) belong to different intervals 𝛥 𝑘,𝑛 , and so 𝑔𝑛 (𝑦) = 𝑚. Similarly, in case 𝐼𝑓 (𝑦) = ∞, an analogous reasoning shows that 𝑔𝑛 (𝑦) → ∞ as 𝑛 → ∞. Moreover, a straightforward calculation shows that ∞

2𝑛

∫ 𝑔𝑛 (𝑦) 𝑑𝑦 = ∑ (𝑀𝑘,𝑛 − 𝑚𝑘,𝑛) , −∞

(1.55)

𝑘=1

where 𝑀𝑘,𝑛 and 𝑚𝑘,𝑛 are given in (1.54). However, the left-hand side of (1.55) tends to the integral of 𝐼𝑓 , by Levi’s theorem (Theorem 0.3), while the right-hand side of (1.55) tends to Var(𝑓; [𝑎, 𝑏]), by (1.39). We will use the second statement of Proposition 1.27 in the proof of an important char­ acterization of absolutely continuous functions (Theorem 3.9). We close this section with another two results on changes of variables for func­ tions of bounded variation. The first result is a characterization of functions of bounded variation due to Federer ([113, Section 2.5.16]) as compositions of mono­ tone and Lipschitz continuous functions with Lipschitz constant¹⁴ 𝐿 = 1. In a certain sense, this characterization may be considered as “dual” to Proposition 1.12. Theorem 1.28. A function 𝑓 belongs to 𝐵𝑉([𝑎, 𝑏]) if and only if it may be represented as composition 𝑓 = 𝑔 ∘ 𝜏, where 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] is increasing and 𝑔 ∈ 𝐿𝑖𝑝([𝑐, 𝑑]) with Lipschitz constant 𝐿 = 1. Proof. Suppose that 𝑓 = 𝑔 ∘ 𝜏, where 𝑔 and 𝜏 have the mentioned properties. Given any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), we get 𝑚

𝑚

Var(𝑓, 𝑃) = ∑ |𝑔(𝜏(𝑡𝑗 )) − 𝑔(𝜏(𝑡𝑗−1 ))| ≤ ∑ |𝜏(𝑡𝑗 ) − 𝜏(𝑡𝑗−1 )| = |𝜏(𝑏) − 𝜏(𝑎)| , 𝑗=1

𝑗=1

and hence 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]). Conversely, let 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), and let 𝜏 := 𝑉𝑓 be the (in­ creasing) variation function (1.13) of 𝑓. We know that 𝜏 maps [𝑎, 𝑏] into [𝑐, 𝑑], where

14 For obvious geometrical reasons, functions with Lipschitz constant 1 are often called nonexpansive in the literature.

82 | 1 Classical BV-spaces 𝑐 = 0 and 𝑑 = Var(𝑓; [𝑎, 𝑏]), but of course 𝜏 need not be onto. If we now define the func­ tion 𝑔 on the range 𝜏([𝑎, 𝑏]) ⊆ [𝑐, 𝑑] by putting 𝑔(𝜏(𝑥)) := 𝑓(𝑥), then the decomposition 𝑓 = 𝑔 ∘ 𝜏 holds by construction. Since |𝑔(𝜏(𝑠)) − 𝑔(𝜏(𝑡))| = |𝑓(𝑠) − 𝑓(𝑡)| ≤ Var(𝑓; [𝑠, 𝑡]) = |𝜏(𝑠) − 𝜏(𝑡)| for 𝑎 ≤ 𝑠 < 𝑡 ≤ 𝑏, the function 𝑔 is Lipschitz continuous with Lipschitz constant 1 on 𝜏([𝑎, 𝑏]) ⊆ [𝑐, 𝑑]. For extending 𝑔 to the whole interval [𝑐, 𝑑], we may now use the McShane exten­ sion described in Theorem 0.42. However, we may construct an extension of 𝑔 from 𝜏([𝑎, 𝑏]) to a Lipschitz continuous map 𝑔 on [𝑐, 𝑑] (even on the whole real line ℝ) in a more explicit way. Indeed, the “convexification” 𝑔 of 𝑔 defined (with 0 ≤ 𝜆 ≤ 1) by {(1 − 𝜆)𝑔(𝑥−) + 𝜆𝑔(𝑥) if 𝑦 = (1 − 𝜆)𝜏(𝑥−) + 𝜆𝜏(𝑥) , 𝑔(𝑦) := { (1 − 𝜆)𝑔(𝑥) + 𝜆𝑔(𝑥+) if 𝑦 = (1 − 𝜆)𝜏(𝑥) + 𝜆𝜏(𝑥+) , { is easily seen to have the same Lipschitz constant as 𝑔, and so we are done.

(1.56)

We will call the decomposition of 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) constructed in Theorem 1.28 the Fed­ erer decomposition in the sequel. The statement of Theorem 1.28 is somewhat surpris­ ing: although functions of bounded variation can have infinitely many discontinuities, these discontinuities may be “smoothed out” after a monotone change of variables by a Lipschitz continuous map. We illustrate this by a simple example involving just one removable discontinuity in the domain of definition 𝑓. Example 1.29. On [𝑎, 𝑏] = [0, 2], consider the characteristic function 𝑓 = 𝜒{1} of the singleton {1}. Let 𝜏 : [0, 2] → [0, 2] be the variation function (1.13) of 𝑓 which, in this case, has the form {0 { { 𝜏(𝑥) = 1 + 𝑠𝑔𝑛(𝑥 − 1) = {1 { { {2

for 0 ≤ 𝑥 < 1 , for 𝑥 = 1 ,

(1.57)

for 1 < 𝑥 ≤ 2.

Observing that 𝜏([0, 2]) = {0, 1, 2} and applying the convexification (1.56) to the function 𝑔 : {0, 1, 2} → ℝ defined by 𝑔(0) = 𝑔(2) = 0 and 𝑔(1) = 1, we end up with the peak function 𝑔 : [0, 2] → ℝ defined by 𝑔(𝑦) := 1 − |𝑦 − 1| which is constructed from 𝑔 in the simplest way by just joining the three points (0, 0), (1, 1) and (2, 0) by straight lines. Clearly, this function is Lipschitz continuous with minimal Lipschitz constant 1 and indeed satisfies 𝑔 ∘ 𝜏 = 𝑔 ∘ 𝜏 = 𝜒{1} . ♥ A parallel result to Theorem 1.28 with 𝑔 ∈ 𝐿𝑖𝑝([𝑎, 𝑏]) replaced by 𝑔 ∈ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) will be given in Theorem 1.41 in the next section. The following result is, in a certain sense, sharper than Theorem 1.28: it shows that continuous functions of bounded variation may be “made” differentiable with bounded derivative (hence, Lipschitz continuous) after a suitable homeomorphic change of variables ([63], see also [64] and [130, Section 3.1]).

1.2 Bounded variation and continuity

| 83

Proposition 1.30. For a function 𝑔 : [𝑎, 𝑏] → ℝ, the following three statements are equivalent. (a) The function 𝑔 is continuous and of bounded variation. (b) There exists a homeomorphism 𝜏 : [𝑎, 𝑏] → [𝑎, 𝑏] such that 𝑔 ∘ 𝜏 : [𝑎, 𝑏] → ℝ is Lipschitz continuous on [𝑎, 𝑏]. (c) There exists a homeomorphism 𝜏 : [𝑎, 𝑏] → [𝑎, 𝑏] such that 𝑔 ∘ 𝜏 : [𝑎, 𝑏] → ℝ is differentiable with a bounded derivative on [𝑎, 𝑏]. Proof. Without loss of generality, we take [𝑎, 𝑏] = [0, 1]. Suppose first that 𝑔 ∈ 𝐶([0, 1]) ∩ 𝐵𝑉([0, 1]) and put Var(𝑔; [0, 1]) =: 𝜔. To prove (b), we define 𝜎 : [0, 1] → [0, 1 + 𝜔] by 𝜎(𝑥) := 𝑥 + 𝑉𝑔 (𝑥) (0 ≤ 𝑥 ≤ 1) , where 𝑉𝑔 denotes the variation function (1.13) of 𝑔. Clearly, 𝜎 is strictly increasing and satisfies 𝜎(0) = 0, 𝜎(1) = 1 + 𝜔, and |𝑔(𝑥) − 𝑔(𝑦)| ≤ |𝑉𝑔 (𝑥) − 𝑉𝑔 (𝑦)| ≤ |𝜎(𝑥) − 𝜎(𝑦)|

(1.58)

for all 𝑥, 𝑦 ∈ [0, 1]. So, the map 𝜏 : [0, 1] → [0, 1] defined by 𝜏(𝑡) := 𝜎−1 (𝑡 + 𝜔𝑡)

(0 ≤ 𝑡 ≤ 1)

(1.59)

is strictly increasing and surjective with 𝜏(0) = 0 and 𝜏(1) = 1, and thus a homeomor­ phism. Moreover, from (1.58), it follows that the map 𝑓 := 𝑔 ∘ 𝜏 satisfies |𝑓(𝑠) − 𝑓(𝑡)| ≤ |𝑉𝑔 (𝜏(𝑠)) − 𝑉𝑔 (𝜏(𝑡))| ≤ |𝜎(𝜏(𝑠)) − 𝜎(𝜏(𝑡))| ≤ (1 + 𝜔)|𝑠 − 𝑡| for all 𝑠, 𝑡 ∈ [0, 1]. This shows that 𝑓 ∈ 𝐿𝑖𝑝([0, 1]) with 𝑙𝑖𝑝(𝑔) ≤ 1 + 𝜔, and so we have proved (b). Now, we suppose that (b) holds and prove (c). We may assume that 𝑔 itself is Lip­ schitz continuous on [0, 1]. In particular, the set of all 𝑡 ∈ [0, 1] at which 𝑔󸀠 (𝑡) does not exist is then a nullset. Choose a 𝐺𝛿 -set 𝐺 ⊇ 𝑁 which is also a nullset. By Zahorski’s theorem [325, 326], we may find a homeomorphism 𝜏 : [0, 1] → [0, 1] which is dif­ ferentiable with bounded derivative 𝜏󸀠 on [0, 1] and satisfies 𝜏󸀠 (𝑡) = 0 precisely for 𝑡 ∈ 𝜏−1 (𝐺). Now, again putting 𝑓 := 𝑔 ∘ 𝜏 and taking 𝑡 ≠ 𝑡0 , the estimate 󵄨󵄨 𝜏(𝑡) − 𝜏(𝑡 ) 󵄨󵄨 󵄨󵄨 𝑓(𝑡) − 𝑓(𝑡 ) 󵄨󵄨 󵄨󵄨 𝑓(𝑡) − 𝑓(𝑡 ) 󵄨󵄨 󵄨󵄨 𝜏(𝑡) − 𝜏(𝑡 ) 󵄨󵄨 󵄨 󵄨 󵄨󵄨 0 󵄨󵄨 0 󵄨󵄨 󵄨󵄨 0 󵄨󵄨 0 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨 󵄨󵄨 ≤ 𝑙𝑖𝑝(𝑔) 󵄨󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑡 − 𝑡0 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜏(𝑡) − 𝜏(𝑡0 ) 󵄨󵄨 󵄨󵄨 𝑡 − 𝑡0 󵄨󵄨 󵄨󵄨 𝑡 − 𝑡0 shows that 𝑓󸀠 (𝑡0 ) = 0 whenever 𝜏(𝑡0 ) ∈ 𝐺. On the other hand, in case 𝜏(𝑡0 ) ∈ ̸ 𝐺, the function 𝑔 is differentiable at 𝑡0 with 𝑓󸀠 (𝑡0 ) = 𝑔󸀠 (𝜏(𝑡0 ))𝜏󸀠 (𝑡0 ),

|𝑓󸀠 (𝑡)| ≤ 𝑙𝑖𝑝(𝑔)‖𝜏󸀠 ‖∞ .

In either case, 𝑓 is differentiable with bounded derivative. Thus, we have proved that (b) implies (c).

84 | 1 Classical BV-spaces The fact that (c) implies (a) follows from Proposition 1.12 since every differentiable map with bounded derivative is Lipschitz continuous, and hence of bounded varia­ tion, and every homeomorphism of an interval onto itself is strictly monotone. Proposition 1.30 (c) gives the best possible change of variables in the following sense: there exist functions 𝑓 ∈ 𝐶([𝑎, 𝑏]) ∩ 𝐵𝑉([𝑎, 𝑏]) such that 𝑓 ∘ 𝜏 is not continuously differ­ entiable for any homeomorphism 𝜏 : [𝑎, 𝑏] → [𝑎, 𝑏], see [130].

1.3 Functions of bounded Wiener variation Now, we consider a certain extension of the spaces 𝐵𝑉([𝑎, 𝑏]) which was introduced in 1924 by Wiener [321]. Definition 1.31. Given a real number 𝑝 ≥ 1, a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), and a function 𝑓 : [𝑎, 𝑏] → ℝ, the nonnegative real number 𝑚

𝑊 𝑝 Var𝑊 𝑝 (𝑓, 𝑃) = Var𝑝 (𝑓, 𝑃; [𝑎, 𝑏]) := ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|

(1.60)

𝑗=1

is called the Wiener variation of 𝑓 on [𝑎, 𝑏] with respect to 𝑃, while the (possibly infi­ nite) number 𝑊 𝑊 Var𝑊 𝑝 (𝑓) = Var𝑝 (𝑓; [𝑎, 𝑏]) := sup {Var𝑝 (𝑓, 𝑃; [𝑎, 𝑏]) : 𝑃 ∈ P([𝑎, 𝑏])} ,

(1.61)

where the supremum is taken over all partitions of [𝑎, 𝑏], is called the total Wiener variation of 𝑓 on [𝑎, 𝑏]. In case Var𝑊 𝑝 (𝑓; [𝑎, 𝑏]) < ∞, we say that 𝑓 has finite Wiener variation on [𝑎, 𝑏] and write¹⁵ 𝑓 ∈ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]). It is useful to complete this definition by defining 𝑊𝐵𝑉∞ ([𝑎, 𝑏]) := 𝑅([𝑎, 𝑏]), that is, the space of regular functions (Section 0.3). As before, we will consider this space endowed with the norm (0.39). ◼ In the following Proposition 1.32, which, to some extent, is parallel to Proposition 1.3, we collect some properties of the quantities (1.60) and (1.61). Proposition 1.32. The quantities (1.60) and (1.61) have the following properties. (a) The 𝑝-th root of the variation (1.61) is subadditive with respect to functions, i.e. 1/𝑝 1/𝑝 1/𝑝 ≤ Var𝑊 + Var𝑊 Var𝑊 𝑝 (𝑓 + 𝑔; [𝑎, 𝑏]) 𝑝 (𝑓; [𝑎, 𝑏]) 𝑝 (𝑔; [𝑎, 𝑏])

(1.62)

for 𝑓, 𝑔 : [𝑎, 𝑏] → ℝ.

15 In the literature, this space is often denoted by 𝐵𝑉𝑝 ([𝑎, 𝑏]). In our notation, the letter 𝑊 stands for “Wiener,” to distinguish this space from another space which we will consider in Definition 2.50 in Chapter 2. Functions 𝑓 ∈ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) are sometimes said to have finite 𝑝-variation in Wiener’s sense.

1.3 Functions of bounded Wiener variation

| 85

(b) The 𝑝-th root of the variation (1.61) is homogeneous with respect to functions, i.e. 1/𝑝 1/𝑝 = |𝜇| Var𝑊 Var𝑊 𝑝 (𝜇𝑓; [𝑎, 𝑏]) 𝑝 (𝑓; [𝑎, 𝑏])

(1.63)

1/𝑝 |𝑓(𝑠) − 𝑓(𝑡)| ≤ Var𝑊 𝑝 (𝑓; [𝑠, 𝑡])

(1.64)

for 𝜇 ∈ ℝ. (c) The estimate holds for 𝑎 ≤ 𝑠 < 𝑡 ≤ 𝑏. (d) Every function 𝑓 ∈ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) is bounded with 1//𝑝 , ‖𝑓‖∞ ≤ |𝑓(𝑎)| + Var𝑊 𝑝 (𝑓; [𝑎, 𝑏])

where the norm ‖ ⋅ ‖∞ is given by (0.39). (e) The linear space 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) equipped with the norm 1/𝑝 ‖𝑓‖𝑊𝐵𝑉𝑝 := |𝑓(𝑎)| + Var𝑊 𝑝 (𝑓; [𝑎, 𝑏])

(1.65)

is a Banach space. Proof. The properties (a)–(d) are proved in exactly the same way as in Proposition 1.3, where, in the proof of (1.62), we use the Minkowski inequality. From (a) and (b), it follows that 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) is a linear space. Moreover, it is not hard to see that (1.65) defines, in fact, a norm on 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]). Let (𝑓𝑛 )𝑛 be a Cauchy se­ quence with respect to this norm. By property (d), (𝑓𝑛 )𝑛 is then also a Cauchy sequence with respect to the norm (0.39), and so there exists a bounded function 𝑓 : [𝑎, 𝑏] → ℝ such that (𝑓𝑛 )𝑛 converges uniformly on [𝑎, 𝑏] to 𝑓. Let 𝜀 > 0, and let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]). If ‖𝑓𝑖 − 𝑓𝑗 ‖𝑊𝐵𝑉𝑝 ≤ 𝜀 for sufficiently large 𝑖 and 𝑗, then 𝑚

𝑝 Var𝑊 𝑝 (𝑓𝑖 − 𝑓, 𝑃; [𝑎, 𝑏]) = ∑ |(𝑓𝑖 − 𝑓)(𝑡𝑘 ) − (𝑓𝑖 − 𝑓)(𝑡𝑘−1 )| 𝑘=1

𝑚

= lim ∑ |(𝑓𝑖 − 𝑓𝑗 )(𝑡𝑘 ) − (𝑓𝑖 − 𝑓𝑗 )(𝑡𝑘−1 )|𝑝 ≤ 𝜀𝑝 , 𝑗→∞

𝑘=1

which together with the trivial fact 𝑓𝑛 (𝑎) → 𝑓(𝑎), implies (e). One might ask why in Proposition 1.32 we did not state analogues to properties (f) (monotonicity with respect to partitions) and (g) (additivity with respect to intervals) of Proposition 1.3. The simple reason is that such analogues are not true, as the fol­ lowing example shows. Example 1.33. Consider the function 𝑓 : [0, 2] → ℝ defined in Example 1.29, i.e. 0 for 0 ≤ 𝑥 < 1 , { { { 𝑓(𝑥) = {1 for 𝑥 = 1 , { { {2 for 1 < 𝑥 ≤ 2,

(1.66)

86 | 1 Classical BV-spaces and consider the partitions 𝑃 := {0, 2} and 𝑄 := {0, 1, 2} ⊃ 𝑃. An easy calculation then shows that 𝑝 Var𝑊 Var𝑊 𝑝 (𝑓, 𝑃; [0, 2]) = 2 , 𝑝 (𝑓, 𝑄; [0, 2]) = 2 , 𝑊 which shows that, in case 𝑝 > 1, 𝑃 ⊆ 𝑄 does not imply Var𝑊 𝑝 (𝑓, 𝑃; [𝑎, 𝑏]) ≤ Var𝑝 (𝑓, 𝑄; [𝑎, 𝑏]), so property (f) of Proposition 1.3 has no analogue. To prove that property (g) of Proposition 1.3 fails, we consider the Wiener variation of the function (1.66) on the intervals [0, 2], [0, 1], and [1, 2]. Similarly, a straightforward computation shows that 𝑊 Var𝑊 𝑝 (𝑓; [0, 1]) = Var𝑝 (𝑓; [1, 2]) = 1 ,

but 𝑝 Var𝑊 𝑝 (𝑓; [0, 2]) = 2 ,

which shows that the Wiener variation is not additive with respect to intervals if 𝑝 > 1, and so Proposition 1.3 (g) has no analogue either. ♥ We remark that for any 𝑝 ∈ [1, ∞), the Wiener variation is superadditive with respect to intervals in the sense that 𝑊 𝑊 Var𝑊 𝑝 (𝑓; [𝑎, 𝑏]) ≥ Var𝑝 (𝑓; [𝑎, 𝑐]) + Var𝑝 (𝑓; [𝑐, 𝑏])

(1.67)

for 𝑎 < 𝑐 < 𝑏. We will prove this in a more general context in Proposition 2.10 (a) in Chapter 2. A comparison of Definition 1.31 with Definition 1.1 shows that Var𝑊 1 (𝑓, 𝑃; [𝑎, 𝑏]) = Var(𝑓, 𝑃; [𝑎, 𝑏]),

Var𝑊 1 (𝑓; [𝑎, 𝑏]) = Var(𝑓; [𝑎, 𝑏]) ,

and so 𝑊𝐵𝑉1 ([𝑎, 𝑏]) = 𝐵𝑉([𝑎, 𝑏]). Moreover, the inclusion 𝐿𝑖𝑝([𝑎, 𝑏]) ⊆ 𝐵𝑉([𝑎, 𝑏]) stated in (1.46) admits the following refinement. Proposition 1.34. For 1 ≤ 𝑝 < ∞, the inclusion 𝐿𝑖𝑝1/𝑝 ([𝑎, 𝑏]) ⊆ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏])

(1.68)

holds. Proof. The proof follows immediately from the estimate 𝑚

𝑚

𝑚

𝑗=1

𝑗=1

𝑗=1

∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|𝑝 ≤ 𝐿𝑝 ∑ |𝑡𝑗 − 𝑡𝑗−1 |𝑝𝛼 = 𝐿𝑝 ∑ |𝑡𝑗 − 𝑡𝑗−1 | = 𝐿𝑝 (𝑏 − 𝑎)

for any function 𝑓 which satisfies (0.67) with 𝛼 = 1/𝑝. The proof of Proposition 1.34 shows that the Banach space (𝐿𝑖𝑝1/𝑝 ([𝑎, 𝑏]), ‖ ⋅ ‖𝐿𝑖𝑝𝛼 ) is continuously imbedded into the Banach space (𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]), ‖⋅‖𝑊𝐵𝑉𝑝 ) with imbedding constant max {1, (𝑏 − 𝑎)1/𝑝 }; compare this with (1.48).

1.3 Functions of bounded Wiener variation

| 87

One could ask whether or not the space 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) is contained in the space 𝑊𝐵𝑉𝑞 ([𝑎, 𝑏]) for certain values of 𝑝 and 𝑞. This is in fact true. Before proving this, we need a series of technical results about convex functions. Let 𝜙 : [0, ∞) → ℝ be a given function, and consider the function 𝜓 : (0, ∞) → ℝ defined by 𝜙(𝑡) (𝑡 > 0). (1.69) 𝜓(𝑡) := 𝑡 Lemma 1.35. If 𝜙 is convex on [0, ∞) and 𝜙(0) ≤ 0, then the function (1.69) is increasing on (0, ∞). Proof. Fix arbitrarily 𝑥1 , 𝑥2 ∈ [0, ∞) with 0 < 𝑥1 < 𝑥2 . Since 𝜙 is convex, we have 𝜙(𝜆𝑠 + (1 − 𝜆)𝑡) ≤ 𝜆𝜙(𝑠) + (1 − 𝜆)𝜙(𝑡) for 𝑠, 𝑡 ∈ [0, ∞) and 𝜆 ∈ [0, 1]. Putting in this inequality 𝑠 = 𝑥2 , 𝑡 = 0, and 𝜆 = 𝑥1 /𝑥2 , and taking into account that 𝜙(0) ≤ 0, we get 𝜙(𝑥1 ) ≤ 𝜆𝜙(𝑥2 ) + (1 − 𝜆)𝜙(0) ≤ 𝜆𝜙(𝑥2 ) =

𝑥1 𝜙(𝑥2 ) . 𝑥2

Consequently, 𝜓(𝑥1 ) =

𝜙(𝑥1 ) 𝜙(𝑥2 ) ≤ = 𝜓(𝑥2 ) 𝑥1 𝑥2

as claimed. In the next lemma, we use the concept of the Young function introduced in Defini­ tion 0.16. Lemma 1.36. Assume that 𝜙 : [0, ∞) → [0, ∞) is a Young function. Then 𝜙 is increasing and superadditive on [0, ∞), i.e. 𝜙(𝛼) + 𝜙(𝛽) ≤ 𝜙(𝛼 + 𝛽)

(𝛼, 𝛽 ≥ 0) .

(1.70)

Proof. Suppose that 𝜙 is not increasing; then, there exist points 𝑥1 , 𝑥2 ∈ [0, ∞) such that 𝑥1 < 𝑥2 and 𝜙(𝑥1 ) > 𝜙(𝑥2 ). If 𝑥1 = 0, then in view of our assumptions, we have 𝜙(𝑥1 ) = 𝜙(0) = 0 < 𝜙(𝑥2 ). Thus, we may assume that 0 < 𝑥1 < 𝑥2 , and hence 1 1 > . 𝑥1 𝑥2 Combining this with our assumption 𝜙(𝑥1 ) > 𝜙(𝑥2 ) > 0, we obtain 𝜙(𝑥1 ) 𝜙(𝑥2 ) 𝜙(𝑥2 ) ≥ > . 𝑥1 𝑥1 𝑥2 However, this contradicts the fact that the function 𝜓 defined in (1.69) is increasing on (0, ∞), as we have proved in Lemma 1.35.

88 | 1 Classical BV-spaces Now, we prove that 𝜙 is superadditive on [0, ∞). Fix 𝛼, 𝛽 ∈ [0, ∞). If 𝛼 = 0, then from our assumptions, we have 𝜙(0) = 0, and so 𝜙(𝛼) + 𝜙(𝛽) = 𝜙(𝛽) = 𝜙(0 + 𝛽) = 𝜙(𝛼 + 𝛽) . The case 𝛽 = 0 may be treated analogously. Therefore, we may suppose without loss of generality that 𝛼 > 0 and 𝛽 > 0, and hence 𝛼 < 𝛼 + 𝛽 and 𝛽 < 𝛼 + 𝛽. By what we just proved, we then get 𝜙(𝛼) 𝜙(𝛼 + 𝛽) ≤ , 𝛼 𝛼+𝛽 and hence 𝜙(𝛼) ≤

𝛼 𝜙(𝛼 + 𝛽), 𝛼+𝛽

𝜙(𝛽) 𝜙(𝛼 + 𝛽) ≤ , 𝛽 𝛼+𝛽 𝛽 𝜙(𝛼 + 𝛽) . 𝛼+𝛽

𝜙(𝛽) ≤

Adding up these inequalities, we obtain (1.70). Example 1.37. The simplest example of a Young function is of course 𝜙(𝑡) = 𝑡𝑝 for 𝑝 ≥ 1. The superadditivity condition (1.70) here simply reads 𝑠𝑝 + 𝑡𝑝 ≤ (𝑠 + 𝑡)𝑝

(𝑠, 𝑡 ≥ 0), ♥

which is of course true if (and only if) 𝑝 ≥ 1.

Now, we are in a position to establish a relation between 𝑊𝐵𝑉𝑝 and 𝑊𝐵𝑉𝑞 for suitable values of 𝑝 and 𝑞. Proposition 1.38. Let 1 ≤ 𝑝 ≤ 𝑞 < ∞. Then the inequality 1/𝑞 1/𝑝 ≤ Var𝑊 Var𝑊 𝑞 (𝑓; [𝑎, 𝑏]) 𝑝 (𝑓; [𝑎, 𝑏])

(1.71)

holds. Consequently, 𝐵𝑉([𝑎, 𝑏]) = 𝑊𝐵𝑉1 ([𝑎, 𝑏]) ⊆ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊆ 𝑊𝐵𝑉𝑞 ([𝑎, 𝑏]) ⊆ 𝐵([𝑎, 𝑏])

(1.72)

for these values of 𝑝 and 𝑞. Proof. Fix an arbitrary partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]). Taking 𝜙(𝑡) := 𝑡𝑞/𝑝 in view of Lemma 1.36 and Example 1.37, we obtain 𝑚

𝑚

𝑗=1

𝑗=1

𝑞/𝑝

∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|𝑞 = ∑ (|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|𝑝 ) 𝑚

𝑞/𝑝 𝑝

≤ (∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| ) 𝑗=1

𝑞/𝑝 ≤ Var𝑊 , 𝑝 (𝑓; [𝑎, 𝑏])

which proves (1.71). The inclusion (1.72) is of course an immediate consequence of (1.71).

1.3 Functions of bounded Wiener variation

|

89

Choosing, in particular, 𝑝 = 1 in (1.72), we see that 𝐵𝑉([𝑎, 𝑏]) ⊆ 𝑊𝐵𝑉𝑞 ([𝑎, 𝑏]) for all 𝑞 ≥ 1. Moreover, (1.71) is 𝑞 Var𝑊 𝑞 (𝑓; [𝑎, 𝑏]) ≤ Var(𝑓; [𝑎, 𝑏])

(𝑓 ∈ 𝐵𝑉([𝑎, 𝑏])),

(1.73)

which may be proved also directly from Hölder’s inequality and shows that the space (𝐵𝑉([𝑎, 𝑏]), ‖ ⋅ ‖𝐵𝑉 ) is imbedded into the space (𝑊𝐵𝑉𝑞 ([𝑎, 𝑏]), ‖ ⋅ ‖𝑊𝐵𝑉𝑞 ) for 𝑞 ≥ 1. If we admit 𝑞 = ∞ in the sense of Definition 1.31, then the inclusion (1.72) still holds true since every function in 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) is regular. At this point, let us collect some of the function classes, together with relations between them, in the following Table 1.1 where 1 < 𝑝 < ∞. Table 1.1. Relations between function classes over 𝐼 = [𝑎, 𝑏]. 𝐿𝑖𝑝1/𝑝 (𝐼) ∪ 𝐿𝑖𝑝(𝐼)

⊂

𝑊𝐵𝑉𝑝 (𝐼)

⊂

𝑅(𝐼)

⊃

⊂

𝐴𝐶(𝐼) ∩ 𝐵𝑉(𝐼) ∩ 𝐼𝑉𝑃(𝐼)

⊂

𝐵𝑉(𝐼) ∩ 𝐶(𝐼) ∪ 𝐶(𝐼)

⊂

𝐵𝑉(𝐼) ∪ 𝐶𝐵𝑉(𝐼)

⊂

𝐼𝑉𝑃(𝐼)

⊂

Later (see Table 2.6 in Chapter 2), we will give an enlarged version of Table 1.1 and show that all inclusions are strict. Let us mention that the inclusion 𝑊𝐵𝑉𝑝 ⊆ 𝑊𝐵𝑉𝑞 proved in Proposition 1.38 is also strict in case 𝑝 < 𝑞. To show this, we consider the zigzag functions 𝑍𝐶,𝐷 introduced in Definition 0.49. It follows from the construction that ∞

𝑝

Var𝑊 𝑝 (𝑍𝐶,𝐷 ; [0, 1]) = ∑ 𝑑𝑘

(1 ≤ 𝑝 < ∞) .

(1.74)

𝑘=1

Observe that the Jordan variation and Wiener variation of 𝑍𝐶,𝐷 is independent of the sequence 𝐶. For the particularly simple example of the special zigzag function (0.93), we get ∞ 1 (𝑍 ; [0, 1]) = ∑ (1 ≤ 𝑝 < ∞) . (1.75) Var𝑊 𝜃 𝑝 𝑝𝜃 𝑘=1 𝑘 This simple observation allows us to show that 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊂ 𝑊𝐵𝑉𝑞 ([𝑎, 𝑏]) for 𝑝 < 𝑞, and even more: Example 1.39. For 𝑝 ≥ 1, consider the function 𝑓(𝑥) := 𝑍1/𝑝 (𝑥). From (1.75), it then follows that 𝑓 ∈ 𝑊𝐵𝑉𝑞 ([0, 1]) \ 𝑊𝐵𝑉𝑝 ([0, 1]) for any 𝑞 > 𝑝. However, we can actually do better: the same function of course satisfies 𝑓 ∈ ⋂ 𝑊𝐵𝑉𝑞 ([0, 1]) \ 𝑊𝐵𝑉𝑝 ([0, 1]) . 𝑞>𝑝

In particular, the special zigzag function 𝑍1 belongs to 𝑊𝐵𝑉𝑝 ([0, 1]) for all 𝑝 > 1, but not to 𝐵𝑉([0, 1]). ♥

90 | 1 Classical BV-spaces We may use the zigzag function 𝑍𝜃 as well to show that the inclusion (1.68) in Propo­ sition 1.34 is strict. Example 1.40. Choosing 𝜃 > 1/𝑝 arbitrary, by (1.75), we conclude that 𝑍𝜃 ∈ 𝑊𝐵𝑉𝑝 ([0, 1]). On the other hand, we have already seen in Corollary 0.51 that no zigzag func­ tion 𝑍𝜃 belongs to the Hölder space 𝐿𝑖𝑝1/𝑝 ([0, 1]). Alternatively, we could have used the function 𝑓 from Example 0.41 which belongs ♥ to 𝑊𝐵𝑉𝑝 ([0, 1]) for any 𝑝 ≥ 1, but not to 𝐿𝑖𝑝𝛼 ([0, 1]) for any 𝛼 ≤ 1. We now consider the Federer decomposition of functions of bounded Wiener varia­ tion. The following is parallel to Theorem 1.28 and may be found in [65] without proof. Yet another generalization of this will be given in Exercise 2.3. Theorem 1.41. A function 𝑓 belongs to 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) if and only if it may be represented as composition 𝑓 = 𝑔 ∘ 𝜏, where 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] is increasing and 𝑔 ∈ 𝐿𝑖𝑝1/𝑝 ([𝑐, 𝑑]) with Hölder constant 𝐿 = 1. Proof. The proof is very similar to that of Theorem 1.28, and thus we only sketch the idea. Suppose that 𝑓 = 𝑔 ∘ 𝜏, where 𝑔 and 𝜏 have the mentioned properties. Given any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), we get 𝑚

𝑝 Var𝑊 𝑝 (𝑓, 𝑃) = ∑ |𝑔(𝜏(𝑡𝑗 )) − 𝑔(𝜏(𝑡𝑗−1 ))| 𝑗=1 𝑚

≤ ∑ |𝜏(𝑡𝑗 ) − 𝜏(𝑡𝑗−1 )| = |𝜏(𝑏) − 𝜏(𝑎)| , 𝑗=1

and hence 𝑓 ∈ 𝐵𝑉𝑝𝑊 ([𝑎, 𝑏]). Conversely, let 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) and let 𝜏(𝑡) := 𝑉𝑓,𝑝 (𝑡), where 𝑉𝑓,𝑝 denotes the variation function 𝑉𝑓,𝑝 (𝑥) := Var𝑊 𝑝 (𝑓; [𝑎, 𝑥])

(𝑎 ≤ 𝑥 ≤ 𝑏) .

Then 𝜏 maps [𝑎, 𝑏] into [𝑐, 𝑑], where 𝑐 = 0 and 𝑑 = Var𝑊 𝑝 (𝑓; [𝑎, 𝑏]). If we define, as in Theorem 1.28, the function 𝑔 on the range 𝜏([𝑎, 𝑏]) ⊆ [𝑐, 𝑑] by putting 𝑔(𝜏(𝑥)) := 𝑓(𝑥), then the decomposition 𝑓 = 𝑔 ∘ 𝜏 holds trivially by construction and 1/𝑝 ≤ |𝜏(𝑠) − 𝜏(𝑡)|1/𝑝 |𝑔(𝜏(𝑠)) − 𝑔(𝜏(𝑡))| = |𝑓(𝑠) − 𝑓(𝑡)| ≤ Var𝑊 𝑝 (𝑓; [𝑠, 𝑡])

for 𝑎 ≤ 𝑠 < 𝑡 ≤ 𝑏, by (1.64), which shows that 𝑔 is Hölder continuous with exponent 𝛼 = 1/𝑝 and Hölder constant 1 on 𝜏([𝑎, 𝑏]). If we extend 𝑔 now from 𝜏([𝑎, 𝑏]) to [𝑐, 𝑑] by using the McShane extension (0.76), we get a function 𝑔̂ ∈ 𝐿𝑖𝑝1/𝑝 ([𝑐, 𝑑]) with Hölder constant 1 such that 𝑓 = 𝑔̂ ∘ 𝜏 = 𝑔 ∘ 𝜏 as claimed. Of course, Theorem 1.28 is contained in Theorem 1.41 in the special case 𝑝 = 1. Also, observe the similarity of these theorems with the Sierpiński decomposition of regular functions stated in Theorem 0.36. We collect the decomposition theorems proved so

1.4 Functions of several variables

| 91

far in the following Table 1.2; here, the function 𝜏 is supposed to map the interval [𝑎, 𝑏] into itself. Table 1.2. Decomposition of functions in the spaces 𝑅, 𝐵𝑉, and 𝑊𝐵𝑉𝑝 . 𝑔∘𝜏 𝑔∘𝜏 𝑔∘𝜏 𝑔∘𝜏

∈ 𝑅([𝑎, 𝑏]) ∈ 𝐵𝑉([𝑎, 𝑏]) ∈ 𝐵𝑉([𝑎, 𝑏]) ∈ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏])

𝑔 ∈ 𝐶([𝑎, 𝑏]) 𝑔 ∈ 𝐿𝑖𝑝([𝑎, 𝑏]) 𝑔 ∈ 𝐵𝑉([𝑎, 𝑏]) 𝑔 ∈ 𝐿𝑖𝑝1/𝑝 ([𝑎, 𝑏])

iff iff iff iff

and and and and

𝜏 strictly monotone 𝜏 monotone 𝜏 pseudomonotone 𝜏 monotone

Table 1.2 shows the subtle, though not surprising, “equilibrium” between the func­ tions 𝜏 and 𝑔: weak properties of 𝑔 have to be “compensated” by strong properties of 𝜏, and vice versa.

1.4 Functions of several variables All functions considered so far were defined on some interval [𝑎, 𝑏]. In view of appli­ cations, however, functions which are defined on the Cartesian product of intervals in higher dimensional Euclidean space are also important. In this section, we discuss several types of variation for such functions and see to what extent our results carry over to higher dimensional domains of definition. For simplicity, we restrict ourselves to rectangles [𝑎, 𝑏] × [𝑐, 𝑑] in the 𝑥𝑦-plane; the generalization to three or more dimen­ sions is straightforward. The most natural approach goes as follows. Given a function 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ and partitions 𝑃 = {𝑠0 , 𝑠1 , . . . , 𝑠𝑚 } ∈ P([𝑎, 𝑏]) and 𝑄 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑛 } ∈ P([𝑐, 𝑑]), consider the three expressions 𝑚

Var(𝑓(⋅, 𝑐), 𝑃; [𝑎, 𝑏]) := ∑ |𝑓(𝑠𝑖 , 𝑐) − 𝑓(𝑠𝑖−1 , 𝑐)| ,

(1.76)

Var(𝑓(𝑎, ⋅), 𝑄; [𝑐, 𝑑]) := ∑ |𝑓(𝑎, 𝑡𝑗 ) − 𝑓(𝑎, 𝑡𝑗−1 )| ,

(1.77)

𝑖=1 𝑛

𝑗=1

and V2 (𝑓, 𝑃 × 𝑄; [𝑎, 𝑏] × [𝑐, 𝑑]) 𝑚

𝑛

:= ∑ ∑ |𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑓(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑓(𝑠𝑖 , 𝑡𝑗−1 ) + 𝑓(𝑠𝑖−1 , 𝑡𝑗−1 )| .

(1.78)

𝑖=1 𝑗=1

Moreover, in analogy to (1.4), Var(𝑓(⋅, 𝑐); [𝑎, 𝑏]) := sup {Var(𝑓(⋅, 𝑐), 𝑃; [𝑎, 𝑏]) : 𝑃 ∈ P([𝑎, 𝑏])} ,

(1.79)

Var(𝑓(𝑎, ⋅); [𝑐, 𝑑]) := sup {Var(𝑓(𝑎, ⋅), 𝑄; [𝑐, 𝑑]) : 𝑄 ∈ P([𝑐, 𝑑])} ,

(1.80)

92 | 1 Classical BV-spaces and V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑])

(1.81)

:= sup {V2 (𝑓, 𝑃 × 𝑄; [𝑎, 𝑏] × [𝑐, 𝑑]) : 𝑃 ∈ P([𝑎, 𝑏]), 𝑄 ∈ P([𝑐, 𝑑])}, where all suprema are taken over the indicated partitions. Definition 1.42. With the above notation, we call the (possibly infinite) number Var(𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) := Var(𝑓(⋅, 𝑐); [𝑎, 𝑏]) + Var(𝑓(𝑎, ⋅); [𝑐, 𝑑]) + V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑])

(1.82)

the total variation of 𝑓 on [𝑎, 𝑏] × [𝑐, 𝑑]. In case Var(𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) < ∞, we say that 𝑓 has bounded variation on [𝑎, 𝑏] × [𝑐, 𝑑] and write 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]). ◼ We now start with a list of properties of the two-dimensional variation (1.82); the fol­ lowing is parallel to (some part of) Proposition 1.3. Proposition 1.43. The quantity (1.82) has the following properties. (a) The variation (1.82) is subadditive with respect to functions, i.e. Var(𝑓 + 𝑔; [𝑎, 𝑏] × [𝑐, 𝑑]) ≤ Var(𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) + Var(𝑔; [𝑎, 𝑏] × [𝑐, 𝑑])

(1.83)

for 𝑓, 𝑔 : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ. (b) The variation (1.82) is homogeneous with respect to functions, i.e. Var(𝜇𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) = |𝜇| Var(𝑓; [𝑎, 𝑏] × [𝑐, 𝑑])

(1.84)

|𝑓(𝑥, 𝑦) − 𝑓(𝜉, 𝜂)| ≤ Var(𝑓; [𝜉, 𝑥] × [𝜂, 𝑦])

(1.85)

for 𝜇 ∈ ℝ. (c) The estimate holds for 𝑎 ≤ 𝜉 < 𝑥 ≤ 𝑏 and 𝑐 ≤ 𝜂 < 𝑦 ≤ 𝑑. (d) Every function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) is bounded with ‖𝑓‖∞ ≤ |𝑓(𝑎, 𝑐)| + Var(𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) ,

(1.86)

‖𝑓‖∞ := sup {|𝑓(𝑥, 𝑦)| : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑐 ≤ 𝑦 ≤ 𝑑} .

(1.87)

where Proof. The assertions (a) and (b) are obvious. To prove (c), fix 𝑎 ≤ 𝜉 < 𝑥 ≤ 𝑏 and 𝑐 ≤ 𝜂 < 𝑦 ≤ 𝑑. Choosing 𝑃 := {𝜉, 𝑥} ∈ P([𝜉, 𝑥]) and 𝑄 := {𝜂, 𝑦} ∈ P([𝜂, 𝑦]), we then have Var(𝑓(⋅, 𝜂), 𝑃; [𝜉, 𝑥]) ≥ |𝑓(𝑥, 𝜂) − 𝑓(𝜉, 𝜂)| , Var(𝑓(𝜉, ⋅), 𝑄; [𝜂, 𝑦]) ≥ |𝑓(𝜉, 𝑦) − 𝑓(𝜉, 𝜂)|

1.4 Functions of several variables

| 93

as well as V2 (𝑓, 𝑃 × 𝑄; [𝑎, 𝑏] × [𝑐, 𝑑]) ≥ |𝑓(𝑥, 𝑦) − 𝑓(𝜉, 𝑦) − 𝑓(𝑥, 𝜂) + 𝑓(𝜉, 𝜂)| . Consequently, |𝑓(𝑥, 𝑦) − 𝑓(𝜉, 𝜂)| = |𝑓(𝑥, 𝜂) − 𝑓(𝜉, 𝜂) + 𝑓(𝜉, 𝑦) − 𝑓(𝜉, 𝜂) + 𝑓(𝑥, 𝑦) − 𝑓(𝜉, 𝑦) − 𝑓(𝑥, 𝜂) + 𝑓(𝜉, 𝜂)| ≤ |𝑓(𝑥, 𝜂) − 𝑓(𝜉, 𝜂)| + |𝑓(𝜉, 𝑦) − 𝑓(𝜉, 𝜂)| + |𝑓(𝑥, 𝑦) − 𝑓(𝜉, 𝑦) − 𝑓(𝑥, 𝜂) + 𝑓(𝜉, 𝜂)| ≤ Var(𝑓(⋅, 𝜂), 𝑃; [𝜉, 𝑥]) + Var(𝑓(𝜉, ⋅), 𝑄; [𝜂, 𝑦]) + V2 (𝑓, 𝑃 × 𝑄; [𝑎, 𝑏] × [𝑐, 𝑑]) = Var(𝑓, [𝜉, 𝑥] × [𝜂, 𝑦]) . Choosing, in particular, (𝜉, 𝜂) := (𝑎, 𝑐) in (c) yields |𝑓(𝑥, 𝑦)| ≤ |𝑓(𝑎, 𝑐)| + Var(𝑓, [𝑎, 𝑥] × [𝑐, 𝑦]) ≤ |𝑓(𝑎, 𝑐)| + Var(𝑓, [𝑎, 𝑏] × [𝑐, 𝑑]) which proves (d) after passing to the supremum over [𝑎, 𝑏] × [𝑐, 𝑑]. We know from Proposition 1.3 (g) that the one-dimensional variations (1.79) and (1.80) are additive with respect to intervals. A similar additivity property holds for the twodimensional variation (1.81): For 𝑎 < 𝑒 < 𝑏 and 𝑐 < 𝑓 < 𝑑, we have V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) = V2 (𝑓; [𝑎, 𝑒] × [𝑐, 𝑓]) + V2 (𝑓; [𝑒, 𝑏] × [𝑐, 𝑓]) + V2 (𝑓; [𝑎, 𝑒] × [𝑓, 𝑑]) + V2 (𝑓; [𝑒, 𝑏] × [𝑓, 𝑑]) .

(1.88)

Proposition 1.43 (d) suggests that one equip the space 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) with the norm (1.89) ‖𝑓‖𝐵𝑉 := |𝑓(𝑎, 𝑐)| + Var(𝑓, [𝑎, 𝑏] × [𝑐, 𝑑]) . In fact, (1.86) shows that with this norm on 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]), the continuous imbedding 𝐵𝑉 󳨅→ 𝐵 holds with imbedding constant 1. Proposition 1.44. The space 𝐵𝑉([𝑎, 𝑏]×[𝑐, 𝑑]) equipped with the norm (1.89) is a Banach algebra satisfying ‖𝑓𝑔‖𝐵𝑉 ≤ 4‖𝑓‖𝐵𝑉 ‖𝑔‖𝐵𝑉 (1.90) for 𝑓, 𝑔 ∈ 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]). Proof. From Proposition 1.43 (a) and (b), it follows that (𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]), ‖ ⋅ ‖𝐵𝑉 ) is a linear space; its completeness is proved exactly as in Proposition 1.10. To prove (1.90), recall that ‖𝑓𝑔‖𝐵𝑉 = |𝑓(𝑎, 𝑐)𝑔(𝑎, 𝑐)| + Var(𝑓(⋅, 𝑐)𝑔(⋅, 𝑐); [𝑎, 𝑏]) + Var(𝑓(𝑎, ⋅)𝑔(𝑎, ⋅); [𝑐, 𝑑]) + V2 (𝑓𝑔; [𝑎, 𝑏] × [𝑐, 𝑑]) .

(1.91)

94 | 1 Classical BV-spaces We estimate the second, third, and fourth term on the right-hand side of (1.91) separately. By (1.17) and (1.11), the second term may be estimated by Var(𝑓(⋅, 𝑐)𝑔(⋅, 𝑐); [𝑎, 𝑏]) ≤ Var(𝑔(⋅, 𝑐); [𝑎, 𝑏]) sup |𝑓(𝑥, 𝑐)| + Var(𝑓(⋅, 𝑐); [𝑎, 𝑏]) sup |𝑔(𝑥, 𝑐)| 𝑎≤𝑥≤𝑏

𝑎≤𝑥≤𝑏

≤ Var(𝑔(⋅, 𝑐); [𝑎, 𝑏]) {|𝑓(𝑎, 𝑐)| + Var(𝑓(⋅, 𝑐); [𝑎, 𝑏])}

(1.92)

+ Var(𝑓(⋅, 𝑐); [𝑎, 𝑏]) {|𝑔(𝑎, 𝑐)| + Var(𝑔(⋅, 𝑐); [𝑎, 𝑏])} = |𝑓(𝑎, 𝑐)| Var(𝑔(⋅, 𝑐); [𝑎, 𝑏]) + |𝑔(𝑎, 𝑐)| Var(𝑓(⋅, 𝑐); [𝑎, 𝑏]) + 2 Var(𝑓(⋅, 𝑐); [𝑎, 𝑏]) Var(𝑔(⋅, 𝑐); [𝑎, 𝑏]). Similarly, for the third term, we get Var(𝑓(𝑎, ⋅)𝑔(𝑎, ⋅); [𝑐, 𝑑]) ≤ |𝑓(𝑎, 𝑐)| Var(𝑔(𝑎, ⋅); [𝑐, 𝑑]) + |𝑔(𝑎, 𝑐)| Var(𝑓(𝑎, ⋅); [𝑐, 𝑑])

(1.93)

+ 2 Var(𝑓(𝑎, ⋅); [𝑐, 𝑑]) Var(𝑔(𝑎, ⋅); [𝑐, 𝑑]). Estimating the fourth term in (1.91) is harder. Fix 𝑃 = {𝑠0 , 𝑠1 , . . . , 𝑠𝑚 } ∈ P([𝑎, 𝑏]) and 𝑄 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑛 } ∈ P([𝑐, 𝑑]). Then for 𝑖 = 1, 2, . . . , 𝑚 and 𝑗 = 1, 2, . . . , 𝑛, we obtain (𝑓𝑔)(𝑠𝑖−1 , 𝑡𝑗−1 ) + (𝑓𝑔)(𝑠𝑖 , 𝑡𝑗 ) − (𝑓𝑔)(𝑠𝑖−1 , 𝑡𝑗 ) − (𝑓𝑔)(𝑠𝑖 , 𝑡𝑗−1 ) = [𝑓(𝑠𝑖−1 , 𝑡𝑗−1 ) + 𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑓(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑓(𝑠𝑖 , 𝑡𝑗−1 )] 𝑔(𝑠𝑖−1 , 𝑡𝑗−1 ) + 𝑓(𝑠𝑖 , 𝑡𝑗 ) [𝑔(𝑠𝑖−1 , 𝑡𝑗−1 ) + 𝑔(𝑠𝑖 , 𝑡𝑗 ) − 𝑔(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑔(𝑠𝑖 , 𝑡𝑗−1 )] + [𝑓(𝑎, 𝑡𝑗 ) − 𝑓(𝑎, 𝑡𝑗−1 )] [𝑔(𝑠𝑖 , 𝑐) − 𝑔(𝑠𝑖−1 , 𝑐)] + [𝑓(𝑎, 𝑡𝑗 ) − 𝑓(𝑎, 𝑡𝑗−1 )] [𝑔(𝑠𝑖−1 , 𝑐) + 𝑔(𝑠𝑖 , 𝑡𝑗−1 ) − 𝑔(𝑠𝑖−1 , 𝑡𝑗−1 ) − 𝑔(𝑠𝑖 , 𝑐)] + [𝑓(𝑎, 𝑡𝑗−1 ) + 𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑓(𝑎, 𝑡𝑗 ) − 𝑓(𝑠𝑖 , 𝑡𝑗−1 )] [𝑔(𝑠𝑖 , 𝑐) − 𝑔(𝑠𝑖−1 , 𝑐)] + [𝑓(𝑎, 𝑡𝑗−1 ) + 𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑓(𝑎, 𝑡𝑗 ) − 𝑓(𝑠𝑖 , 𝑡𝑗−1 )] × [𝑔(𝑠𝑖−1 , 𝑐) + 𝑔(𝑠𝑖 , 𝑡𝑗−1 ) − 𝑔(𝑠𝑖−1 , 𝑡𝑗−1 ) − 𝑔(𝑠𝑖 , 𝑐)] + [𝑓(𝑠𝑖 , 𝑐)) − 𝑓(𝑠𝑖−1 , 𝑐)] [𝑔(𝑎, 𝑡𝑗 ) − 𝑔(𝑎, 𝑡𝑗−1 )] + [𝑓(𝑠𝑖 , 𝑐) + 𝑓(𝑠𝑖−1 , 𝑐)] [𝑔(𝑎, 𝑠𝑗−1 ) + 𝑔(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑔(𝑎, 𝑡𝑗 ) − 𝑔(𝑠𝑖−1 , 𝑡𝑗−1 )] + [𝑓(𝑠𝑖−1 , 𝑐)) + 𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑓(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑓(𝑠𝑖 , 𝑐))] [𝑔(𝑎, 𝑡𝑗 ) − 𝑔(𝑎, 𝑡𝑗−1 )] + [𝑓(𝑠𝑖−1 , 𝑐) + 𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑓(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑓(𝑠𝑖 , 𝑐)] × [𝑔(𝑎, 𝑡𝑗−1 ) + 𝑔(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑔(𝑎, 𝑡𝑗 ) − 𝑔(𝑠𝑖−1 , 𝑡𝑗−1 )] =: 𝐴 𝑖𝑗 + 𝐵𝑖𝑗 + 𝐶𝑖𝑗 + 𝐷𝑖𝑗 + 𝐸𝑖𝑗 + 𝐹𝑖𝑗 + 𝐺𝑖𝑗 + 𝐻𝑖𝑗 + 𝐼𝑖𝑗 + 𝐽𝑖𝑗 . We estimate the 10 terms 𝐴 𝑖𝑗 , . . . , 𝐽𝑖𝑗 separately, summing over 𝑖 = 1, . . . , 𝑚 and 𝑗 = 1, . . . , 𝑛. By (1.87), we have 𝑚 𝑛 𝑚 𝑛 󵄨 󵄨󵄨 󵄨 ∑ ∑ |𝐴 𝑖𝑗 | = ∑ ∑ 󵄨󵄨󵄨󵄨𝑓(𝑎, 𝑡𝑗 ) − 𝑓(𝑎, 𝑡𝑗−1 )󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑔(𝑠𝑖 , 𝑐) − 𝑔(𝑠𝑖−1 , 𝑐)󵄨󵄨󵄨 𝑖=1 𝑗=1

𝑖=1 𝑗=1

≤ V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑])‖𝑔‖∞

1.4 Functions of several variables

| 95

and 𝑚

𝑛

∑ ∑ |𝐵𝑖𝑗 | 𝑖=1 𝑗=1

𝑚 𝑛 󵄨 󵄨 = ∑ ∑ 𝑓(𝑠𝑖 , 𝑡𝑗 ) 󵄨󵄨󵄨󵄨𝑔(𝑠𝑖−1 , 𝑡𝑗−1 ) + 𝑔(𝑠𝑖 , 𝑡𝑗 ) − 𝑔(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑔(𝑠𝑖 , 𝑡𝑗−1 )󵄨󵄨󵄨󵄨 𝑖=1 𝑗=1

≤ V2 (𝑔; [𝑎, 𝑏] × [𝑐, 𝑑])‖𝑓‖∞ . Clearly, 𝑚

𝑛

𝑚

𝑛

󵄨 󵄨󵄨 󵄨 ∑ ∑ |𝐶𝑖𝑗 | = ∑ ∑ 󵄨󵄨󵄨󵄨𝑓(𝑎, 𝑡𝑗 ) − 𝑓(𝑎, 𝑡𝑗−1 )󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑔(𝑠𝑖 , 𝑐) − 𝑔(𝑠𝑖−1 , 𝑐)󵄨󵄨󵄨 𝑖=1 𝑗=1

𝑖=1 𝑗=1

≤ Var(𝑓(𝑎, ⋅); [𝑐, 𝑑]) Var(𝑔(⋅, 𝑐); [𝑎, 𝑏]) and, by symmetry, 𝑚 𝑛 𝑚 𝑛 󵄨 󵄨󵄨 󵄨 ∑ ∑ |𝐺𝑖𝑗 | = ∑ ∑ 󵄨󵄨󵄨𝑓(𝑠𝑖 , 𝑐)) − 𝑓(𝑠𝑖−1 , 𝑐)󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑔(𝑎, 𝑡𝑗 ) − 𝑔(𝑎, 𝑡𝑗−1 )󵄨󵄨󵄨󵄨 𝑖=1 𝑗=1

𝑖=1 𝑗=1

≤ Var(𝑓(⋅, 𝑐); [𝑎, 𝑏]) Var(𝑔(𝑎, ⋅); [𝑐, 𝑑]) . Now, from the additivity property (1.88) it follows that 𝑚

𝑛

V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) = ∑ ∑ V2 (𝑓; [𝑠𝑖−1 , 𝑠𝑖 ] × [𝑡𝑗−1 , 𝑡𝑗 ]) , 𝑖=1 𝑗=1

and similarly for 𝑔, i.e. 𝑚

𝑛

V2 (𝑔; [𝑎, 𝑏] × [𝑐, 𝑑]) = ∑ ∑ V2 (𝑔; [𝑠𝑖−1 , 𝑠𝑖 ] × [𝑡𝑗−1 , 𝑡𝑗 ]) . 𝑖=1 𝑗=1

From this, we conclude that 𝑚

𝑛

∑ ∑ |𝐷𝑖𝑗 | 𝑖=1 𝑗=1

𝑚 𝑛 󵄨 󵄨󵄨 󵄨 = ∑ ∑ 󵄨󵄨󵄨󵄨𝑓(𝑎, 𝑡𝑗 ) − 𝑓(𝑎, 𝑡𝑗−1 )󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑔(𝑠𝑖−1 , 𝑐) + 𝑔(𝑠𝑖 , 𝑡𝑗−1 ) − 𝑔(𝑠𝑖−1 , 𝑡𝑗−1 ) − 𝑔(𝑠𝑖 , 𝑐)󵄨󵄨󵄨󵄨 𝑖=1 𝑗=1

≤ Var(𝑓(𝑎, ⋅); [𝑐, 𝑑]) V2 (𝑔; [𝑎, 𝑏] × [𝑐, 𝑑]) , and 𝑚

𝑛

∑ ∑ |𝐸𝑖𝑗 | 𝑖=1 𝑗=1

𝑚 𝑛 󵄨 󵄨󵄨 󵄨 = ∑ ∑ 󵄨󵄨󵄨󵄨𝑓(𝑎, 𝑡𝑗−1 ) + 𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑓(𝑎, 𝑡𝑗 ) − 𝑓(𝑠𝑖 , 𝑡𝑗−1 )󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑔(𝑠𝑖 , 𝑐) − 𝑔(𝑠𝑖−1 , 𝑐)󵄨󵄨󵄨 𝑖=1 𝑗=1

≤ Var(𝑔(⋅, 𝑐); [𝑎, 𝑏]) V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑])

96 | 1 Classical BV-spaces as well as 𝑚

𝑛

∑ ∑ |𝐻𝑖𝑗 | 𝑖=1 𝑗=1

𝑚 𝑛 󵄨 󵄨 󵄨󵄨 = ∑ ∑ 󵄨󵄨󵄨𝑓(𝑠𝑖 , 𝑐) + 𝑓(𝑠𝑖−1 , 𝑐)󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑔(𝑎, 𝑠𝑗−1 ) + 𝑔(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑔(𝑎, 𝑡𝑗 ) − 𝑔(𝑠𝑖−1 , 𝑡𝑗−1 )󵄨󵄨󵄨󵄨 𝑖=1 𝑗=1

≤ Var(𝑓(⋅, 𝑐); [𝑎, 𝑏]) V2 (𝑔; [𝑎, 𝑏] × [𝑐, 𝑑]) , and 𝑚

𝑛

∑ ∑ |𝐼𝑖𝑗 | 𝑖=1 𝑗=1

𝑚 𝑛 󵄨 󵄨󵄨 󵄨 = ∑ ∑ 󵄨󵄨󵄨󵄨𝑓(𝑠𝑖−1 , 𝑐)) + 𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑓(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑓(𝑠𝑖 , 𝑐))󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑔(𝑎, 𝑡𝑗 ) − 𝑔(𝑎, 𝑡𝑗−1 )󵄨󵄨󵄨󵄨 𝑖=1 𝑗=1

≤ Var(𝑔(𝑎, ⋅); [𝑐, 𝑑]) V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) . It remains to estimate the long terms 𝐹𝑖𝑗 and 𝐽𝑖𝑗 . Here, we get 𝑚 𝑛 𝑚 𝑛 󵄨 󵄨 ∑ ∑ |𝐹𝑖𝑗 | = ∑ ∑ 󵄨󵄨󵄨󵄨𝑓(𝑎, 𝑡𝑗−1 ) + 𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑓(𝑎, 𝑡𝑗 ) − 𝑓(𝑠𝑖 , 𝑡𝑗−1 )󵄨󵄨󵄨󵄨 𝑖=1 𝑗=1

𝑖=1 𝑗=1

󵄨 󵄨 × 󵄨󵄨󵄨󵄨𝑔(𝑠𝑖−1 , 𝑐) + 𝑔(𝑠𝑖 , 𝑡𝑗−1 ) − 𝑔(𝑠𝑖−1 , 𝑡𝑗−1 ) − 𝑔(𝑠𝑖 , 𝑐)󵄨󵄨󵄨󵄨 ≤ V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) V2 (𝑔; [𝑎, 𝑏] × [𝑐, 𝑑]) , and, by symmetry, 𝑚 𝑛 𝑚 𝑛 󵄨 󵄨 ∑ ∑ |𝐽𝑖𝑗 | = ∑ ∑ 󵄨󵄨󵄨󵄨𝑓(𝑠𝑖−1 , 𝑐) + 𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑓(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑓(𝑠𝑖 , 𝑐)󵄨󵄨󵄨󵄨 𝑖=1 𝑗=1

𝑖=1 𝑗=1

󵄨 󵄨 × 󵄨󵄨󵄨󵄨𝑔(𝑎, 𝑡𝑗−1 ) + 𝑔(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑔(𝑎, 𝑡𝑗 ) − 𝑔(𝑠𝑖−1 , 𝑡𝑗−1 )󵄨󵄨󵄨󵄨 ≤ V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) V2 (𝑔; [𝑎, 𝑏] × [𝑐, 𝑑]) . Adding up all these estimates, we obtain V2 (𝑓𝑔; [𝑎, 𝑏] × [𝑐, 𝑑]) ≤ |𝑓(𝑎, 𝑐)| V2 (𝑔; [𝑎, 𝑏] × [𝑐, 𝑑]) + 2 Var(𝑓(⋅, 𝑐); [𝑎, 𝑏]) V2 (𝑔; [𝑎, 𝑏] × [𝑐, 𝑑]) + 2 Var(𝑓(𝑎, ⋅); [𝑐, 𝑑]) V2 (𝑔; [𝑎, 𝑏] × [𝑐, 𝑑]) + V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑])|𝑔(𝑎, 𝑐)| + 2 V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) Var(𝑔(⋅, 𝑐); [𝑎, 𝑏]) + 2 V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) Var(𝑔(𝑎, ⋅); [𝑐, 𝑑]) + Var(𝑓(⋅, 𝑐); [𝑎, 𝑏]) Var(𝑔(𝑎, ⋅); [𝑐, 𝑑]) + Var(𝑓(𝑎, ⋅); [𝑐, 𝑑]) Var(𝑔(⋅, 𝑐); [𝑎, 𝑏]) + 4 V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) . Finally, combining this with (1.91), (1.92) and (1.93), we arrive at (1.90).

1.4 Functions of several variables

| 97

Applying Proposition 0.31 (or Exercise 0.30) to (1.90), we see that replacing (1.89) by the norm (1.94) ⦀𝑓⦀𝐵𝑉 := ‖𝑓‖∞ + Var(𝑓, [𝑎, 𝑏] × [𝑐, 𝑑]) with ‖𝑓‖∞ given by (1.87), we get ‖𝑓‖𝐵𝑉 ≤ ⦀𝑓⦀𝐵𝑉 ≤ 4‖𝑓‖𝐵𝑉 (i.e. the norms (1.89) and (1.94) are equivalent), and (1.90) may be strengthened to ⦀𝑓𝑔⦀𝐵𝑉 ≤ ⦀𝑓⦀𝐵𝑉 ⦀𝑔⦀𝐵𝑉

(1.95)

which means that (𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]), ⦀ ⋅ ⦀𝐵𝑉 ) is a normalized Banach algebra. Our discussion shows that Definition 1.42 is quite natural. This definition goes back to Hardy and Krause.¹⁶ However, there are several other definitions which go back to Vitali, Fréchet, Arzelà, and Tonelli¹⁷ and are worth being mentioned. The cor­ responding definitions for functions 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ read as follows. Definition 1.45 (Vitali). A function 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ has bounded variation in Vitali’s sense if V 2 (𝑓; [𝑎, 𝑏]×[𝑐, 𝑑]) < ∞, see (1.81). In this case, we write 𝑓 ∈ 𝑉𝐵𝑉([𝑎, 𝑏]× [𝑐, 𝑑]). ◼ Definition 1.46 (Fréchet). Given a function 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ and partitions 𝑃 = {𝑠0 , 𝑠1 , . . . , 𝑠𝑚 } ∈ P([𝑎, 𝑏]) and 𝑄 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑛 } ∈ P([𝑐, 𝑑]), consider the expression V±2 (𝑓, 𝑃 × 𝑄; [𝑎, 𝑏] × [𝑐, 𝑑]) 𝑚

𝑛

:= ∑ ∑ 𝜖𝑖 𝜖𝑗 [𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑓(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑓(𝑠𝑖 , 𝑡𝑗−1 ) + 𝑓(𝑠𝑖−1 , 𝑡𝑗−1 )] ,

(1.96)

𝑖=1 𝑗=1

where 𝜖𝑖 , 𝜖𝑗 ∈ {−1, 1}. If V±2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) := sup {V±2 (𝑓, 𝑃 × 𝑄; [𝑎, 𝑏] × [𝑐, 𝑑]) : 𝑃 ∈ P([𝑎, 𝑏]), 𝑄 ∈ P([𝑐, 𝑑])} < ∞ , we write 𝑓 ∈ 𝐹𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) and say that 𝑓 has bounded variation in Fréchet’s sense. ◼ Definition 1.47 (Arzelà). Given a function 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ and partitions¹⁸ 𝑃 = {𝑠0 , 𝑠1 , . . . , 𝑠𝑚 } ∈ P([𝑎, 𝑏]) and 𝑄 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑐, 𝑑]), consider the expression 𝑚 𝑚

Var𝐴 (𝑓, 𝑃 × 𝑄; [𝑎, 𝑏] × [𝑐, 𝑑]) := ∑ ∑ |𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑓(𝑠𝑖−1 , 𝑡𝑗−1 )| .

(1.97)

𝑖=1 𝑗=1

16 Functions in 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) in the sense of Hardy and Krause have been studied in detail in Hildebrandt’s book [147]. 17 We do not cite the original papers of these authors here, but refer the reader to the books [139] or [148]. 18 Here, it is important that both partitions contain the same number of points.

98 | 1 Classical BV-spaces In case Var𝐴 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) := sup {Var𝐴 (𝑓, 𝑃 × 𝑄; [𝑎, 𝑏] × [𝑐, 𝑑]) : 𝑃 ∈ P([𝑎, 𝑏]), 𝑄 ∈ P([𝑐, 𝑑])} < ∞ , we write 𝑓 ∈ 𝐴𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) and say that 𝑓 has bounded variation in Arzelà’s sense. ◼ Definition 1.48 (Tonelli). A function 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] has bounded variation in Tonelli’s sense if Var(𝑓(⋅, 𝑦); [𝑎, 𝑏]) < ∞ for almost all 𝑦 ∈ [𝑐, 𝑑], see (1.79), Var(𝑓(𝑥, ⋅); [𝑐, 𝑑]) < ∞ for almost all 𝑥 ∈ [𝑎, 𝑏], see (1.80) and, in addition, 𝑏

∫ Var(𝑓(𝑥, ⋅); [𝑐, 𝑑]) 𝑑𝑥 < ∞

(1.98)

𝑎

and

𝑑

∫ Var(𝑓(⋅, 𝑦); [𝑎, 𝑏]) 𝑑𝑦 < ∞.

(1.99)

𝑐

In this case, we write 𝑓 ∈ 𝑇𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]).

◼

Of course, it is interesting to establish some interconnections between these concepts of bounded variation. First of all, it follows directly from the definitions that 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) ⊆ 𝑉𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) ⊆ 𝐹𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) ,

(1.100)

𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) ⊆ 𝐴𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) ,

(1.101)

𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) ⊆ 𝑇𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) .

(1.102)

and

Therefore, the question arises if there are other inclusions, or even some equality, between these classes, and if the inclusions in (1.100)–(1.102) are strict. These ques­ tions have been discussed in the survey paper [100], where the authors consider the following examples over the unit square 𝑆 := [0, 1] × [0, 1]. Example 1.49. Let 𝐷 := {(𝑥, 𝑦) : 0 ≤ 𝑦 ≤ 𝑥 ≤ 1} and 𝑓 := 𝜒𝐷 . Then 𝑓 ∈ 𝑇𝐵𝑉(𝑆), but 𝑓 ∈ ̸ 𝐴𝐵𝑉(𝑆) ∪ 𝑉𝐵𝑉(𝑆). ♥ Example 1.50. Let 𝑓 : 𝑆 → ℝ be defined by¹⁹ {𝑥 sin 𝑥1 𝑓(𝑥, 𝑦) := { 0 { Then 𝑓 ∈ 𝐴𝐵𝑉(𝑆) ∩ 𝑉𝐵𝑉(𝑆), but 𝑓 ∈ ̸ 𝐵𝑉(𝑆).

19 So, 𝑓 does not depend on the second variable.

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 . ♥

1.4 Functions of several variables

| 99

Example 1.51. Let 𝑀 ⊂ [0, 1] be a nonmeasurable set, 𝐷 := {(𝑥, −𝑥) : 𝑥 ∈ 𝑀}, and ♥ 𝑓 := 𝜒𝐷 . Then 𝑓 ∈ 𝐴𝐵𝑉(𝑆), but 𝑓 ∈ ̸ 𝑇𝐵𝑉(𝑆). Example 1.52. Let 𝑆 be divided into quarter squares, and let 𝑆1 be the upper left-hand quarter square, i.e. 𝑆1 = [0, 1/2] × [1/2, 1]. Next, divide the lower right-hand square into quarter squares, and let 𝑆2 be that quarter which has a common vertex with 𝑆1 , i.e. 𝑆2 = [1/2, 3/4] × [1/4, 1/2]. Continuing this process, we obtain an infinite sequence (𝑆𝑛)𝑛 of square subdivisions of 𝑆 converging toward the point (1, 0). On the square 𝑆1 , we construct a “point-rectangle function” 𝑓 satisfying V2 (𝑓; 𝑆1 ) = 1,

V±2 (𝑓; 𝑆1 ) ≤

1 . 2

Similarly, on the square 𝑆𝑘 , we construct 𝑓 as a “point-rectangle function” satis­ fying 1 V2 (𝑓; 𝑆𝑘 ) = 1, V±2 (𝑓; 𝑆𝑘 ) ≤ 𝑘 . 2 Putting 𝑓(𝑥, 𝑦) := 0 at all remaining points, we see that 𝑓 ∈ 𝐹𝐵𝑉(𝑆), but 𝑓 ∈ ̸ 𝑉𝐵𝑉(𝑆). ♥ Example 1.53. Let 𝐷 := {(𝑥, −𝑥) : 0 ≤ 𝑥 ≤ 1} and 𝑓 := 𝜒𝐷 . Then 𝑓 ∈ 𝐴𝐵𝑉(𝑆), but 𝑓 ∈ ̸ 𝐹𝐵𝑉(𝑆). ♥ We summarize our results in the following Table 1.3. It is clear that no entries are pos­ sible in the diagonal and the first column because the inclusions (1.100)–(1.102) show that 𝐵𝑉 is the smallest of all considered spaces.²⁰ Moreover, the second inclusion in (1.100) implies that we cannot find a function which belongs to 𝑉𝐵𝑉, but not to 𝐹𝐵𝑉. Table 1.3. 𝐵𝑉-functions over [𝑎, 𝑏] × [𝑐, 𝑑]. Function 𝑓

∈ 𝐵𝑉

∈ 𝑉𝐵𝑉

∈ 𝐹𝐵𝑉

∈ 𝐴𝐵𝑉

∈ 𝑇𝐵𝑉

∉ 𝐵𝑉 ∉ 𝑉𝐵𝑉 ∉ 𝐹𝐵𝑉 ∉ 𝐴𝐵𝑉 ∉ 𝑇𝐵𝑉

– – – – –

Example 1.50 – – Example 1.50 Example 1.50

Example 1.52 Example 1.52 – Example 1.50 Example 1.50

Example 1.50 Example 1.53 Example 1.53 – Example 1.51

Example 1.49 Example 1.49 Example 1.53 Example 1.49 –

In the paper [100], the authors also consider intersections of the five classes occurring in Table 1.3. For example, it is not hard to show that 𝑉𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) ∩ 𝐴𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) = 𝑉𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) ∩ 𝑇𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) = 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) .

20 This explains why functions from 𝐵𝑉 have nicer properties than those from the other spaces.

100 | 1 Classical BV-spaces On the other hand, Example 1.52 shows that the inclusion 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) ⊂ 𝑉𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) ∩ 𝑇𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) is strict. Other functions of this type are given in Exercises 1.61–1.64. We also remark that continuous functions in the classes 𝑉𝐵𝑉, 𝐹𝐵𝑉, 𝐴𝐵𝑉, and 𝑇𝐵𝑉 have been studied in [100] and by other authors. For example, an example of a function 𝑓 ∈ (𝐶∩𝐴𝐵𝑉)\𝐵𝑉 may be found in [174, 175].

1.5 Comments on Chapter 1 As mentioned at the beginning, Jordan’s pioneering paper [153] may be considered as a starting point of the study of functions of bounded variation. Jordan also introduced the variation function (1.13) and used it to prove the decomposition theorem (Theo­ rem 1.5) for 𝐵𝑉-functions. The functions 𝑝𝑓 and 𝑛𝑓 used in the proof of Theorem 1.6 are sometimes called the positive variation and negative variation of 𝑓, respectively. As Theorem 1.26 shows, the variation function 𝑉𝑓 inherits many important prop­ erties from its parent function 𝑓, see [149, 281]. However, Theorem 1.26 (d) also shows that, in contrast to Lipschitz continuity, Hölder continuity is not perfectly symmetric. We state this as Problem 1.1. Does 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) (0 < 𝛼 < 1) imply that 𝑉𝑓 ∈ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏])? By formula (1.17), 𝐵𝑉([𝑎, 𝑏]) is an algebra. Apparently, the idea to use the decomposi­ tion from Theorem 1.6 to show that 𝐵𝑉([𝑎, 𝑏]) is even a normalized algebra, see (1.18), is first given by Kuller in his book [173]. For the space 𝐵𝑉𝑜 ([𝑎, 𝑏]), a similar proof was later given by Bullen [73] and Russell [282] who seemed to have been unaware of Kuller’s proof. Functions of bounded variation are “intermediate” between step functions and regular functions in the sense that 𝑆([𝑎, 𝑏]) ⊆ 𝐵𝑉([𝑎, 𝑏]) ⊆ 𝑅([𝑎, 𝑏]) .

(1.103)

Although both inclusions are actually strict, the three sets in (1.103) are not “too distant” from each other; in fact, in Proposition 0.57, we have shown that the closure of the left set (in the norm of 𝐵([𝑎, 𝑏])) coincides with the right set. A refinement of this may be found in Exercise 1.32. Eduard Helly was an Austrian mathematician whose work had a profound in­ fluence on Riesz and Banach. His famous selection principle [144] (Theorem 1.11) is also known as Helly’s second theorem.²¹ Helly’s selection principle may be viewed as a

21 There is another result known as Helly’s first theorem which refers to Riemann–Stieltjes integrals, see Theorem 4.21 (c) in Chapter 4.

1.5 Comments on Chapter 1 | 101

certain counterpart of the Arzelà–Ascoli criterion (Proposition 0.55) for 𝐵𝑉-functions. More information on Helly’s theorem and its history can be found in the books [76, 182, 238]. The harmless looking Proposition 1.12 has interesting consequences. Consider, for example, the strictly increasing homeomorphism 𝜏(𝑡) := 𝑡𝛾 on [0, 1], where 0 < 𝛾 < 1. The oscillation function 𝑔 = 𝑓𝛼,𝛽 defined in (0.86) belongs to 𝐵𝑉([0, 1]) if either 𝛽 > 0 and 𝛼+𝛽 ≥ 0, or 𝛽 ≤ 0 and 𝛼+𝛽 > 0, see Exercises 1.8 and 1.9. In case 𝛼+𝛽 ≥ 1, we even have 𝑔 ∈ 𝐿𝑖𝑝([0, 1]), as Proposition 0.48 (b) shows, and so we may use the inclusions (1.46) to deduce that also 𝑔 ∈ 𝐵𝑉([0, 1]). Now, for the function 𝑓 = 𝑔 ∘ 𝜏 = 𝑓𝛼𝛾,𝛽𝛾 , we get the same conditions for 𝑓 ∈ 𝐵𝑉([0, 1]), as one could expect from Proposition 1.12. On the other hand, 𝑓 ∈ ̸ 𝐿𝑖𝑝([0, 1]) if 𝛾 is so small that (𝛼 + 𝛽)𝛾 < 1, and so we cannot use (1.46) in this case. Our discussion of the relation of the space 𝐵𝑉 with the class 𝐼𝑉𝑃 in Section 1.2 has been inspired by the paper [179]. The following proposition is a straightforward extension of Proposition 1.12 to functions of bounded Wiener variation in Wiener’s sense. The proof is the same and may be found in [232]. Proposition 1.54. Given a function 𝑔 : [𝑐, 𝑑] → ℝ, let 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] be continuous and strictly increasing with 𝜏(𝑎) = 𝑐 and 𝜏(𝑏) = 𝑑. Then 𝑔 ∘ 𝜏 ∈ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) if and only if 𝑔 ∈ 𝑊𝐵𝑉𝑝 ([𝑐, 𝑑]). Example 1.13 shows again that we cannot drop the continuity assumption on 𝜏 in Proposition 1.54. The following example shows that we cannot drop the monotonicity assumption either. Example 1.55. For 𝑝 ≥ 1, define 𝜏 : [0, 1] → [0, 1] by 󵄨𝑝 󵄨 {𝑡 󵄨󵄨󵄨sin 1𝑡 󵄨󵄨󵄨 󵄨 𝜏(𝑡) := { 󵄨 0 {

for 0 < 𝑡 ≤ 1 , for 𝑡 = 0 .

Then 𝜏 is continuous, but of course far from being monotone. The function 𝑔 : [0, 1] → ℝ defined by 𝑔(𝑥) := 𝑥1/𝑝 belongs to 𝐿𝑖𝑝1/𝑝 ([0, 1]), and hence to 𝑊𝐵𝑉𝑝 ([0, 1]), by (1.68). On the other hand, the function 𝑓 = 𝑔 ∘ 𝜏 does not belong to 𝑊𝐵𝑉𝑝 ([0, 1]), which can be seen by a similar reasoning as in Example 1.8. For 𝑛 ∈ ℕ, consider the partition 𝑃𝑛 := {0, 1} ∪ {𝑠1 , . . . , 𝑠𝑛 } ∪ {𝑡1 , . . . , 𝑡𝑛 } , where 𝑠𝑗 :=

1 , 4𝑗𝜋

𝑡𝑗 :=

1 (4𝑗 + 1)𝜋

(𝑗 = 1, 2, . . . , 𝑛) .

Since 𝑔(𝜏(𝑠𝑗 )) = 0 and 𝑔(𝜏(𝑡𝑗 )) = 𝑡𝑗 , the partition 𝑃𝑛 gives the contribution 1 2 1/𝑝 𝑛 Var𝑊 ∑ , 𝑝 (𝑓, 𝑃𝑛 ; [0, 1]) ≥ ( ) 𝜋 (4𝑘 + 1)1/𝑝 𝑘=1

(1.104)

102 | 1 Classical BV-spaces and the sum in (1.104) is unbounded as 𝑛 → ∞ because 𝑝 ≥ 1.

♥

What we call pseudomonotone functions (Definition 1.14) was introduced and dis­ cussed by Josephy in [155] (under a different name). Proposition 1.15 shows that the class of pseudomonotone functions is situated between monotone and 𝐵𝑉-functions, while Proposition 1.17 shows that pseudomonotone functions provide the “tailormade” substitutions for preserving bounded variation. In Section 1.2, we have discussed several interconnections between bounded vari­ ation and continuity. Several of our examples in this section are taken from, or at least inspired by, Chapters 14 and 15 of the excellent book [39]. The functions having the intermediate value property are interesting from an analytical viewpoint, but have “bad” algebraic properties. For example, it is not hard to see that 𝐼𝑉𝑃([𝑎, 𝑏]) is not a linear space. In fact, the two (discontinuous) functions {sin 𝑓(𝑥) := { 1 {

1 𝑥

for 𝑥 ≠ 0 for 𝑥 = 0

and {sin 1 for 𝑥 ≠ 0 𝑔(𝑥) := { 𝑥 0 for 𝑥 = 0 { both belong to 𝐼𝑉𝑃([−1, 1]), say, but their difference 𝑓−𝑔 = 𝜒{0} does not. So, one could ask how the smallest vector space 𝑋 = span 𝐼𝑉𝑃([𝑎, 𝑏]) looks like. Since 𝐼𝑉𝑃([𝑎, 𝑏]) is one of the largest function classes in Table 1.1, one could expect that also 𝑋 is quite large. In fact, the answer is as simple as surprising: 𝑋 is the space of all functions on [𝑎, 𝑏] ([268, Theorem 9.5])! Proposition 1.30 shows that a functions 𝑔 ∈ 𝐶([𝑎, 𝑏]) ∩ 𝐵𝑉([𝑎, 𝑏]) can be trans­ formed by a suitable homeomorphism 𝜏 : [𝑎, 𝑏] → [𝑎, 𝑏] into a differentiable function 𝑔 ∘ 𝜏 with bounded derivative. There is a (partial) analogue to Proposition 1.30 for functions 𝑔 ∈ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) which reads as follows: Proposition 1.56. For a function 𝑔 : [𝑎, 𝑏] → ℝ, the following two statements are equiv­ alent. (a) The function 𝑔 is continuous and of bounded Wiener 𝑝-variation. (b) There exists a homeomorphism 𝜏 : [𝑎, 𝑏] → [𝑎, 𝑏] such that 𝑔 ∘ 𝜏 : [𝑎, 𝑏] → ℝ is Hölder continuous on [𝑎, 𝑏] with Hölder exponent 1/𝑝. The proof of this proposition is very similar to that of Proposition 1.30 and may be found in [232]. The reader should observe the subtle difference between Proposi­ tion 1.54 and Proposition 1.56: while a function 𝑔 ∈ 𝑊𝐵𝑉𝑝 , in general, remains in 𝑊𝐵𝑉𝑝 after a homeomorphic change of variable, a continuous function 𝑔 ∈ 𝑊𝐵𝑉𝑝 may even become Hölder continuous of order 1/𝑝. So, adding continuity bridges the gap (which is essential, as Example 1.40 shows) between 𝐿𝑖𝑝1/𝑝 and 𝑊𝐵𝑉𝑝 .

1.5 Comments on Chapter 1

| 103

Functions of bounded Wiener variation are briefly treated in the Lecture Notes [65], a good survey article is [117]. A comparison of (1.46) and (1.68), or Theorem 1.28 and Theorem 1.41, shows that 𝑊𝐵𝑉𝑝 -functions are related to Hölder continuous func­ tions in rather the same way as 𝐵𝑉-functions to Lipschitz continuous functions. Such relations are discussed in the recent paper [232]. Less is known for 𝐵𝑉-functions 𝑓 : [𝑎, 𝑏] → 𝑋, where 𝑋 is a normed or even metric space. A good survey on results of this type, including extensions of Proposition 1.7, Theorem 1.11 and Theorem 1.28 to this more general setting is [85]. If 𝑋 is a normed space and Var(𝑓; [𝑎, 𝑏], 𝑋) is defined in the obvious way, in [85], it is also shown that 𝑏

Var(𝑓; [𝑎, 𝑏], 𝑋) ≤ ∫ ‖𝑓󸀠 (𝑥)‖𝑋 𝑑𝑥

(1.105)

𝑎

for 𝑓 ∈ 𝐶1 ([𝑎, 𝑏], 𝑋). We will discuss further results of this type for 𝐵𝑉-functions and absolutely continuous functions in Chapter 3. In the special case 𝑋 = ℝ𝑛 with the Euclidean norm ‖(𝜉1 , 𝜉2 , . . . , 𝜉𝑛)‖ = √𝜉12 + 𝜉22 + ⋅ ⋅ ⋅ + 𝜉𝑛2 , the corresponding space 𝐵𝑉([𝑎, 𝑏], ℝ𝑛 ) has been studied by many authors; in this case, we have equality in (1.105) for 𝑓 ∈ 𝐶1 ([𝑎, 𝑏], ℝ𝑛 ). A particularly important case is 𝑛 = 2; here, we get 𝑏 2

Var(𝑓; [𝑎, 𝑏], ℝ ) = ∫ √𝑓1󸀠 (𝑥)2 + 𝑓2󸀠 (𝑥)2 𝑑𝑥

(1.106)

𝑎

for 𝑓 ∈ 𝐶1 ([𝑎, 𝑏], ℝ2 ), where 𝑓1 and 𝑓2 are the components of 𝑓. We will come back to this special case in connection with rectifiable graphs in Section 3.4. Set-valued maps of bounded variation (in the classical or more general sense) have been studied by many authors, among them [88, 222, 223, 231, 292, 327]. 𝐵𝑉-functions of several (in particular, two) variables on Cartesian products of intervals (in particular, rectangles in the plane) have been studied extensively by Chistyakov [90–95]. Specifically, in [90], the author studies the Banach algebra 𝐵𝑉([𝑎, 𝑏]×[𝑐, 𝑑]) which we considered in Section 1.4, while 𝐵𝑉-functions of 𝑛 variables on a product [𝑎1 , 𝑏1 ] × [𝑎2 , 𝑏2 ] × ⋅ ⋅ ⋅ × [𝑎𝑛 , 𝑏𝑛 ] ⊂ ℝ𝑛 are discussed in [92, 93]. Functions of two variables which belong to the space 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏] × [𝑐, 𝑑]), i.e. have bounded Wiener 𝑝-variation on a rectangle in the plane, are considered in [96]. An analogue of Helly’s selection principle (Theorem 1.11) for functions of several variables may be found in [178]. The other concepts of variation for functions of two variables given in Defini­ tions 1.45–1.48 and Examples 1.49–1.53 are discussed in detail in [100], see also [3, 4]. These concepts are obviously motivated by various applications which we will, in part, consider in subsequent chapters. Our original Definition 1.42 singles out a class of functions arising in the theory of (double) Fourier series. Vitali’s Definition 1.45 is

104 | 1 Classical BV-spaces sufficient to insure the existence of the Riemann–Stieltjes (double) integral for contin­ uous functions of two variables. In particular, if such a function is of the form 𝑓(𝑥, 𝑦) = 𝑔(𝑥)ℎ(𝑦), Fréchet’s concept (which is weaker than Vitali’s concept) is suitable. Finally, Arzelà’s Definition 1.47 is modeled after the decomposition of a 𝐵𝑉-function as the dif­ ference of two monotonically increasing functions. However, among all of these con­ cepts, our Definition 1.42 is by far the most natural one.²² We remark that there are also some papers on 𝐵𝑉-functions with weight, e.g. [54]. Needless to say, the concept of variation has been extended in many directions, lead­ ing sometimes to straightforward generalizations, and sometimes to unexpected new phenomena. Many such extensions will be discussed in detail in the next chapter.

1.6 Exercises to Chapter 1 We state some exercises on the topics covered in this chapter; exercises marked with an asterisk * are more difficult. Exercise 1.1. Let 𝑓, 𝑔 ∈ 𝐵𝑉([𝑎, 𝑏]) with |𝑔(𝑥)| ≥ 𝑐 for some 𝑐 > 0, i.e. 𝑔 is bounded away from zero. Show that 𝑓/𝑔 ∈ 𝐵𝑉([𝑎, 𝑏]). Exercise 1.2. Find functions 𝑓, 𝑔 ∈ 𝐵𝑉([0, 1]) such that 𝑔(𝑥) > 0 on [0, 1] and 𝑓/𝑔 ∈ ̸ 𝐵𝑉([0, 1]). Exercise 1.3. Show that 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) implies |𝑓| ∈ 𝐵𝑉([𝑎, 𝑏]) and Var(|𝑓|; [𝑎, 𝑏]) ≤ Var(𝑓; [𝑎, 𝑏]). Exercise 1.4. Find a function 𝑓 ∈ ̸ 𝐵𝑉([0, 1]) such that |𝑓| ∈ 𝐵𝑉([0, 1]). Exercise 1.5. Suppose that 𝑓 ∈ 𝐶([𝑎, 𝑏]) and |𝑓| ∈ 𝐵𝑉([𝑎, 𝑏]). Show that 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), and compare this with your example in Exercise 1.4. Exercise 1.6. Given 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), show that 󵄨 󵄨 𝑉𝑓 (𝑥0 +) − 𝑉𝑓 (𝑥0 ) = 󵄨󵄨󵄨𝑓(𝑥0 +) − 𝑓(𝑥0 )󵄨󵄨󵄨 for each 𝑥0 ∈ [𝑎, 𝑏), and 󵄨 󵄨 𝑉𝑓 (𝑥0 ) − 𝑉𝑓 (𝑥0 −) = 󵄨󵄨󵄨𝑓(𝑥0 ) − 𝑓(𝑥0 −)󵄨󵄨󵄨 for each 𝑥0 ∈ (𝑎, 𝑏]. Use this to give an alternative proof of Proposition 1.7. Exercise 1.7. Let 𝑓 ∨ 𝑔 and 𝑓 ∧ 𝑔 be defined as in Exercise 0.70. Show that 𝑓, 𝑔 ∈ 𝐵𝑉([𝑎, 𝑏]) implies 𝑓∨𝑔 ∈ 𝐵𝑉([𝑎, 𝑏]) and 𝑓∧𝑔 ∈ 𝐵𝑉([𝑎, 𝑏]), and express Var(𝑓∨𝑔; [𝑎, 𝑏]) and Var(𝑓 ∧ 𝑔; [𝑎, 𝑏]) through Var(𝑓; [𝑎, 𝑏]) and Var(𝑔; [𝑎, 𝑏]).

22 Another reason which makes this transparent will be contained in Theorem 3.51 in Chapter 3.

1.6 Exercises to Chapter 1

| 105

Exercise 1.8. Let 𝛼 ∈ ℝ and 𝛽 > 0. Show that the function (0.86) then belongs to 𝐵𝑉([0, 1]) if and only if 𝛼 + 𝛽 ≥ 0, and calculate 𝑉𝑓 in this case. Exercise 1.9. Let 𝛼 ∈ ℝ and 𝛽 ≤ 0. Show that the function (0.86) then belongs to 𝐵𝑉([0, 1]) if and only if 𝛼 + 𝛽 > 0, and calculate 𝑉𝑓 in this case. Compare this with Example 1.8. Exercise 1.10. Show that a function 𝑓 : ℝ → ℝ is monotone if and only if 𝑓−1 ([𝛼, 𝛽]) is an interval for each interval [𝛼, 𝛽] ⊆ ℝ. Exercise 1.11. Obviously, the set of all monotone functions on [𝑎, 𝑏] contains a one-di­ mensional linear space (namely, all constant functions), as well as a two-dimensional linear space (namely, all affine functions). Does it also contain a three-dimensional linear space? Exercise 1.12. Prove that a function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) has, at most, countably many points of discontinuity, all of them being of first kind (jumps) or removable. Exercise 1.13. Prove the following converse of Exercise 12: given a countable set 𝐷 ⊂ [𝑎, 𝑏], there exists a monotone function 𝑓 such that 𝐷(𝑓) = 𝐷, see (0.49). In particular, construct such a function for 𝐷 := [𝑎, 𝑏] ∩ ℚ. Exercise 1.14*. Construct a function 𝑓 ∈ 𝐵([0, 1]) \ 𝐵𝑉([0, 1]) and a sequence (𝑃𝑛 )𝑛 of partitions 𝑃𝑛 ∈ P([0, 1]) such that Var(𝑓, 𝑃𝑛 ; [0, 1]) = 0 for all 𝑛 ∈ ℕ and 𝜇(𝑃𝑛 ) → 0 as 𝑛→∞. Exercise 1.15. Show that the variation function (1.13) of the function 𝑓(𝑥) = sin 𝑥 on [0, 2𝜋] has the form for 0 ≤ 𝑥 ≤ 12 𝜋 ,

sin 𝑥 { { { 𝑉𝑓 (𝑥) = {2 − sin 𝑥 { { {4 + sin 𝑥

for 12 𝜋 < 𝑥 ≤ 32 𝜋 , for 32 𝜋 < 𝑥 ≤ 2𝜋 .

Exercise 1.16. Show that a function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) and its variation function (1.13) satisfy the inequality 𝑏

𝑏

𝑏

1 (∫ 𝑉𝑓 (𝑥) 𝑑𝑥) (∫ 𝑓(𝑥) 𝑑𝑥) ∫ 𝑉𝑓 (𝑥)𝑓(𝑥) 𝑑𝑥 − 𝑏−𝑎 𝑎

𝑎

𝑏

≤ ∫ 𝑉𝑓 (𝑥)2 𝑑𝑥 − 𝑎

𝑎

𝑏

2

1 (∫ 𝑉𝑓 (𝑥) 𝑑𝑥) . 𝑏−𝑎 𝑎

Illustrate this by means of a (nonmonotone) function 𝑓 of your choice. Exercise 1.17. Let (𝑓𝑛 )𝑛 be a sequence of functions which converges pointwise on [𝑎, 𝑏] to some function 𝑓. If each 𝑓𝑛 is increasing, show that 𝑓 is also increasing. If each 𝑓𝑛 is of bounded variation, does it follow that 𝑓 is of bounded variation?

106 | 1 Classical BV-spaces Exercise 1.18. Let 𝑓, 𝑔 ∈ 𝐵𝑉([𝑎, 𝑏]) such that |𝑓(𝑥)| ≤ |𝑔(𝑥)|. Does it follow that Var(𝑓; [𝑎, 𝑏]) ≤ Var(𝑔; [𝑎, 𝑏])? Exercise 1.19. Construct an example of a sequence in 𝐵𝑉([𝑎, 𝑏]) which is bounded with respect to the norm (1.16), but contains no subsequence, which is convergent in this norm. Compare with Proposition 1.9. Exercise 1.20*. Let 𝑀 ⊂ 𝑁𝐵𝑉([𝑎, 𝑏]) (Definition 1.2) be bounded and closed. Solve Exercise 13.34 in [108] which gives a necessary and sufficient condition for the com­ pactness of 𝑀 in (𝐵𝑉([𝑎, 𝑏]), ‖ ⋅ ‖𝐵𝑉 ). Exercise 1.21*. Show that 𝐵𝑉([𝑎, 𝑏]) may be decomposed as direct sum 𝐵𝑉([𝑎, 𝑏]) = 𝑁𝐵𝑉([𝑎, 𝑏]) ⊕ 𝑉𝐵𝑉([𝑎, 𝑏]) , where 𝑉𝐵𝑉([𝑎, 𝑏]) consists of all functions in 𝐵𝑉([𝑎, 𝑏]) which vanish, except for a countable set of points. Prove that 𝑉𝐵𝑉([𝑎, 𝑏]) is isometrically isomorphic to a certain 𝐿 1 -space. Use this fact and Exercise 1.20 to give a compactness criterion in the space 𝐵𝑉([𝑎, 𝑏]). Exercise 1.22. Using Exercise 1.20, but not general facts from functional analysis, show that the closed unit ball in (𝐵𝑉([𝑎, 𝑏]), ‖ ⋅ ‖𝐵𝑉 ) is not compact. Exercise 1.23. The Hellinger integral of a function 𝑓 : [𝑎, 𝑏] → ℝ is defined as 𝑏

{𝑚 } ∫(𝑑𝑓)2 := sup { ∑[𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )]2 : {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏])} , 𝑎 {𝑗=1 } where the supremum is taken over all partitions of [𝑎, 𝑏]. Show that if this integral exists, then 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) and 𝑏

Var(𝑓; [𝑎, 𝑏]) ≤ √ (𝑏 − 𝑎) ∫(𝑑𝑓)2 . 𝑎

Is the converse also true, i.e. does every function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) have a Hellinger inte­ gral? Exercise 1.24. With the definition and notation of Exercise 1.23, prove that the Hellinger integral for 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) exists if and only if the Hellinger integral for its variation function 𝑉𝑓 exists. Moreover, show that in this case, both integrals coin­ cide. Exercise 1.25. Suppose that the Hellinger integral (Exercise 1.23) for 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) exists. Prove that then both 𝑓 ∈ 𝐿𝑖𝑝1/2 ([𝑎, 𝑏]) and 𝑉𝑓 ∈ 𝐿𝑖𝑝1/2 ([𝑎, 𝑏]) with 𝑏

𝑙𝑖𝑝1/2 (𝑓) ≤ 𝑙𝑖𝑝1/2 (𝑉𝑓 ) ≤ √ ∫(𝑑𝑓)2 , 𝑎

1.6 Exercises to Chapter 1 | 107

where 𝑙𝑖𝑝𝛼 (𝑓) is defined by (0.69). Exercise 1.26. Determine all 𝛼, 𝛽 ∈ ℝ for which the function (0.86) has a Hellinger integral. Compare with Exercises 1.8, 1.9 and 0.52. Exercise 1.27. Given 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), consider the functions 𝑝𝑓 and 𝑛𝑓 constructed in the proof of Theorem 1.6. Prove that 𝑝𝑓 and 𝑛𝑓 are the “most slowly increasing func­ tions” among all Jordan decompositions of 𝑓 in the following sense: If 𝜙, 𝜓 : [𝑎, 𝑏] → ℝ are monotonically increasing such that 𝑓 = 𝜙 − 𝜓, then 𝑝𝑓 (𝑦) − 𝑝𝑓 (𝑥) ≤ 𝜙(𝑦) − 𝜙(𝑥) and 𝑛𝑓 (𝑦) − 𝑛𝑓 (𝑥) ≤ 𝜓(𝑦) − 𝜓(𝑥) for all 𝑥, 𝑦 such that 𝑎 ≤ 𝑥 < 𝑦 ≤ 𝑏. Also, show that Var(𝑓; [𝑥, 𝑦]) = Var(𝑝𝑓 ; [𝑥, 𝑦]) + Var(𝑛𝑓 ; [𝑥, 𝑦]) for 𝑎 ≤ 𝑥 < 𝑦 ≤ 𝑏. Exercise 1.28. Show that every monotone function is Riemann integrable on each compact interval. Deduce that the same is true for 𝐵𝑉-functions. Exercise 1.29*. Prove that every monotone function is a.e. differentiable. Deduce that the same is true for 𝐵𝑉-functions. Exercise 1.30*. Given 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), show that 𝜔1 (𝑓, [𝑎, 𝑏]; 𝛿) = 𝑂(𝛿)

(𝛿 → 0+) ,

where 𝜔𝑝 (𝑓, [𝑎, 𝑏]; 𝛿) denotes the integral modulus of continuity (0.98). Compare with Proposition 0.53 and Exercise 0.76. Exercise 1.31*. Prove the following converse of Exercise 1.30: if 𝑓 ∈ 𝐿 1 ([𝑎, 𝑏]) satisfies 𝜔1 (𝑓, [𝑎, 𝑏]; 𝛿) = 𝑂(𝛿)

(𝛿 → 0+) ,

then 𝑓 is equivalent to some 𝐵𝑉-function on [𝑎, 𝑏]. Exercise 1.32*. Prove that the following three statements for a function 𝑓 : [𝑎, 𝑏] → ℝ are equivalent: (a) 𝑓 is the uniform limit of a sequence of step functions. (b) 𝑓 is the uniform limit of a sequence of 𝐵𝑉-functions. (c) 𝑓 is regular, i.e. the unilateral limits (0.54) exist for all 𝑥0 ∈ [𝑎, 𝑏]. Exercise 1.33. Let 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]). Deduce from Proposition 1.7 that 𝑓 may be represented as the difference of two continuous increasing functions on [𝑎, 𝑏]. Is the same true for functions 𝑓 ∈ 𝐶𝐵𝑉([𝑎, 𝑏])?

108 | 1 Classical BV-spaces Exercise 1.34. Let {𝑥1 , 𝑥2 , 𝑥3 , . . . } ⊂ [𝑎, 𝑏] be the set of discontinuities of a function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) (Exercise 1.12). The jump function 𝜎𝑓 : [𝑎, 𝑏] → ℝ of 𝑓 is then defined by 𝜎𝑓 (𝑎) := 0 and 𝜎𝑓 (𝑡) := ∑ (𝑓(𝑥𝑘 +) − 𝑓(𝑥𝑘 −)) + (𝑓(𝑡) − 𝑓(𝑡−))

(𝑎 < 𝑡 ≤ 𝑏) .

𝑡>𝑥𝑘

Show that 𝑛

∑ (|𝑓(𝑥𝑘 +) − 𝑓(𝑥𝑘 )| + |𝑓(𝑥𝑘 ) − 𝑓(𝑥𝑘 −)|) ≤ Var(𝑓; [𝑎, 𝑏]) 𝑘=1

for all 𝑛 ∈ ℕ. Conclude from this that the jump function 𝜎𝑓 is well-defined. Exercise 1.35. Let 𝜎𝑓 be the jump function of a function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) (Exercise 1.34). Prove that 𝑓 − 𝜎𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]). Also, calculate 𝜎𝑓 for 𝑓 as in Example 1.4. Exercise 1.36. Let 𝑓 : [𝑎, 𝑏] → ℝ be increasing, and let 𝜎𝑓 be the jump function of a function 𝑓 (Exercise 1.34). Show that 𝑓 − 𝜎𝑓 is also increasing. Exercise 1.37. Let 𝜎𝑓 be the jump function of a function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) (Exercise 1.34). Show that Var(𝑓; [𝑎, 𝑏]) = Var(𝑓 − 𝜎𝑓 ; [𝑎, 𝑏]) + Var(𝜎𝑓 ; [𝑎, 𝑏]). Exercise 1.38. Let 𝑓 : [𝑎, 𝑏] → ℝ be a bounded function and 𝑎 < 𝑐 < 𝑏. Suppose that 𝑓 has local bounded variation near 𝑐, which means that there exists a 𝛿 > 0 such that 𝑓 ∈ 𝐵𝑉([𝑐 − 𝛿, 𝑐 + 𝛿]). Also, assume that 𝑓 has a primitive on both (𝑎, 𝑐) and (𝑐, 𝑏), i.e. 𝑓 = 𝐹1󸀠 for some differentiable function 𝐹1 : (𝑎, 𝑐) → ℝ and 𝑓 = 𝐹2󸀠 for some differentiable function 𝐹2 : (𝑐, 𝑏) → ℝ. Show that 𝑓 has a primitive on (𝑎, 𝑏) if and only if 𝑓 is continuous at 𝑐. Exercise 1.39. Given a continuous function 𝑓 : [𝑎, 𝑏] → ℝ and a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), let osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) be defined as in (1.12). Show that 𝑚

Var(𝑓, 𝑃; [𝑎, 𝑏]) ≤ ∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) ≤ Var(𝑓; [𝑎, 𝑏]) 𝑗=1

and conclude that this implies (1.39). Also, illustrate (1.39) by means of a function 𝑓 ∈ 𝐶([𝑎, 𝑏]) ∩ 𝐵𝑉([𝑎, 𝑏]) and a func­ tion 𝑓 ∈ 𝐶([𝑎, 𝑏]) \ 𝐵𝑉([𝑎, 𝑏]) (e.g. the function from Example 1.8).

1.6 Exercises to Chapter 1 |

109

Exercise 1.40. Calculate the functions 𝜏 and 𝑔 from Theorem 1.28 for 𝑓 = 𝑓𝛼,𝛽 with 𝛼 and 𝛽 as in Exercises 1.8 and 1.9. Exercise 1.41. Suppose that 𝑓 ∈ 𝐶([𝑎, 𝑏]) is injective, and 𝑔 : [𝑎, 𝑏] → ℝ has a prim­ itive. Show that the product 𝑓𝑔 has a primitive. See what happens if you drop one of the hypotheses on 𝑓. Exercise 1.42. Suppose that a function 𝑓 : [𝑎, 𝑏] → ℝ has the intermediate value property, and that |𝑓| ∈ 𝐵𝑉([𝑎, 𝑏]). Prove that 𝑓 is continuous, and compare with Ex­ ercises 1.4 and 1.5. Exercise 1.43. Suppose that a function 𝑓 : [𝑎, 𝑏] → [𝑎, 𝑏] has the intermediate value property, and that 𝑓 ∘ 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]). Prove that 𝑓 ∘ 𝑓 is continuous. Is 𝑓 itself then also continuous? Exercise 1.44. Suppose that 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]), and 𝑔 : [𝑎, 𝑏] → ℝ has a primi­ tive. Show that the product 𝑓𝑔 has a primitive. Check what happens if you drop one of the hypotheses on 𝑓. Exercise 1.45. Suppose that 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) has the property that there is a function 𝑔 : [𝑎, 𝑏] → ℝ \ {0} such that both 𝑔 and the product 𝑓𝑔 have a primitive. Prove that the product 𝑓ℎ then has a primitive for any function ℎ : [𝑎, 𝑏] → ℝ which has a primitive. See what happens if you drop one of the hypotheses on 𝑓. Exercise 1.46. Calculate the Sierpiński decomposition (Theorem 0.36) for the func­ tion 𝑓 in Example 1.25. Exercise 1.47. Let 𝐼𝑓 be the Banach indicatrix (Definition 0.38) of the function (1.14). Show by a direct calculation (i.e. without using Proposition 1.27) that 𝐼𝑓 ∈ ̸ 𝐿 1 (ℝ). Exercise 1.48. Calculate the McShane extension (0.76) of the function 𝑔 in Exam­ ple 1.24, and compare the result with the convexification 𝑔 which we have constructed in Theorem 1.28. Exercise 1.49. By considering the characteristic functions 𝜒{𝑐} for 𝑎 ≤ 𝑐 ≤ 𝑏, show that the space 𝐵𝑉([𝑎, 𝑏]) with norm (1.16) is not separable, i.e. there is no countable dense subset. Is the space 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]) with norm (1.16) separable? Exercise 1.50. Let 𝑓 : [−1, 1] → ℝ be defined by 𝑓(𝑥) := |𝑥|. Prove that 𝑉𝑓 ∈ 𝐶1 ([−1, 1]), although 𝑓 is not differentiable at 0. Exercise 1.51. Let 𝑓 := 𝑓2,𝛽 : [−1, 1] → ℝ be the oscillation function from (0.86), where −2 < 𝛽 < −1. Prove that 𝑉𝑓 ∈ ̸ 𝐶1 ([−1, 1]), although 𝑓 is differentiable at 0. Exercise 1.52*. Given 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), prove that 𝑉𝑓󸀠 (𝑥) = |𝑓󸀠 (𝑥)| a.e. on [𝑎, 𝑏].

110 | 1 Classical BV-spaces Exercise 1.53*. Given 𝜙 ∈ 𝐵𝑉([𝑎, 𝑏]) and 𝜓 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]), denote by supp 𝜓 the set of all points 𝑥 ∈ (𝑎, 𝑏) such that 𝜓(𝑥) ≠ 0. Denoting further by (𝑎𝑘 , 𝑏𝑘 ) the open intervals between two subsequent zeros of 𝜓, i.e. ∞

supp 𝜓 ⊆ ⋃(𝑎𝑘 , 𝑏𝑘 ) , 𝑘=1

prove that ∞

Var(𝜙𝜓; [𝑎, 𝑏]) ≤ ‖𝜙‖∞ Var(𝜓; [𝑎, 𝑏]) + ∑ Var(𝜙𝜓; [𝑎𝑘 , 𝑏𝑘 ]) . 𝑘=1

Exercise 1.54. Illustrate Exercise 1.53 by choosing for 𝜙 and 𝜓 suitable oscillation functions of type (0.86). Exercise 1.55*. Suppose that 𝑓 : [𝑎, 𝑏] → ℝ is monotone with 𝐷(𝑓) = [𝑎, 𝑏] ∩ ℚ. Prove that there is no interval [𝑐, 𝑑] ⊆ [𝑎, 𝑏] such that 𝑓 ∈ 𝐼𝑉𝑃([𝑐, 𝑑]). Check this result for the function you constructed in Exercise 1.13. Exercise 1.56. A function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) is said to have vanishing variation at 𝑡0 ∈ [𝑎, 𝑏] if lim Var(𝐹𝑡0 ,𝛿 ; [𝑎, 𝑏]) = 0 , 𝛿→0

where 𝐹𝑡0 ,𝛿 : [𝑎, 𝑏] → ℝ is defined by {𝑓(𝑡) − 𝑓(𝑡0 ) if |𝑡 − 𝑡0 | ≤ 𝛿 , 𝐹𝑡0 ,𝛿 (𝑡) := { 0 otherwise. { Show that 𝑓 has vanishing variation at 𝑡0 if 𝑓 is continuous at 𝑡0 . Exercise 1.57. Show that the function (0.86) belongs to 𝑊𝐵𝑉𝑝 ([0, 1]) for fixed 𝑝 ∈ [1, ∞) if and only if either 𝛽 > 0 and 𝑝𝛼 + 𝛽 ≥ 0, or 𝛽 ≤ 0 and 𝑝𝛼 + 𝛽 > 0. Com­ pare this with Exercises 1.8 and 1.9. Exercise 1.58. Use the result of Exercise 1.57 to construct a function 𝑓 ∈ 𝑊𝐵𝑉𝑝 ([0, 1])\ 𝐿𝑖𝑝1/𝑝 ([0, 1]). Exercise 1.59. Prove the following generalization of Proposition 1.34. For 1 ≤ 𝑝 < ∞, let 𝜔(𝑡) = 𝑂(𝑡1/𝑝 ) as 𝑡 → 0+, where 𝜔 is an arbitrary modulus of continuity; then, 𝐿𝑖𝑝𝜔,∞ ([𝑎, 𝑏]) ⊆ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]), where the space 𝐿𝑖𝑝𝜔,∞ ([𝑎, 𝑏]) is introduced in Defini­ tion 0.54. Exercise 1.60*. The following result shows that the statement of Exercise 1.59 is sharp. For 1 ≤ 𝑝 < ∞, suppose that 𝜔(𝑡) ≠ 𝑂(𝑡1/𝑝 ), but 𝑡1/𝑝 = 𝑜(𝜔(𝑡)) as 𝑡 → 0+, i.e. lim

𝑡→0+

𝜔(𝑡) = ∞. 𝑡1/𝑝

Prove that then there exists a function 𝑓 ∈ 𝐿𝑖𝑝𝜔,∞ ([0, 1]) \ 𝑊𝐵𝑉𝑝 ([0, 1]).

1.6 Exercises to Chapter 1 |

111

Exercise 1.61. Let the rational points of [0, 1] be enumerated, i.e. [0, 1] ∩ ℚ = {𝑟1 , 𝑟2 , 𝑟3 , . . . }, and define a function 𝑓 : [0, 1] × [0, 1] → ℝ by {1 𝑓(𝑥, 𝑦) := { 0 {

if 𝑥 = 𝑟𝑘 , 𝑦 ∈ ̸ ℚ, and 𝑦 > 1 − 1/2𝑘 , otherwise.

Check whether or not 𝑓 belongs to the space 𝐵𝑉([0, 1] × [0, 1]) introduced in Defi­ nition 1.42. Exercise 1.62. Check whether or not the function 𝑓 from Exercise 0.61 belongs to one of the spaces 𝑉𝐵𝑉([0, 1] × [0, 1]), 𝐹𝐵𝑉([0, 1] × [0, 1]), 𝐴𝐵𝑉([0, 1] × [0, 1]), or 𝑇𝐵𝑉([0, 1] × [0, 1]) introduced in Definitions 1.45–1.48. Exercise 1.63. Define a function 𝑓 : [0, 1] × [0, 1] → ℝ by {1 𝑓(𝑥, 𝑦) := { 0 {

if 𝑥 ∈ ℚ and 𝑦 ∈ ℚ , otherwise.

Check whether or not 𝑓 belongs to the space 𝐵𝑉([0, 1] × [0, 1]) introduced in Defini­ tion 1.42. Exercise 1.64. Check whether or not the function 𝑓 from Exercise 0.63 belongs to one of the spaces 𝑉𝐵𝑉([0, 1] × [0, 1]), 𝐹𝐵𝑉([0, 1] × [0, 1]), 𝐴𝐵𝑉([0, 1] × [0, 1]), or 𝑇𝐵𝑉([0, 1] × [0, 1]) introduced in Definitions 1.45–1.48. Exercise 1.65. Show that both inclusions in (1.103) are strict.

2 Nonclassical BV-spaces The function space 𝐵𝑉([𝑎, 𝑏]) discussed in the previous chapter has been generalized in various directions. N. Wiener and L. C. Young distorted the measurement of inter­ vals in the range of functions by considering 𝑝-th powers or, more generally, contin­ uous increasing gauge functions 𝜙. Subsequently, D. Waterman and M. Schramm ad­ mitted countable families of such gauge functions in order to generalize the concept of variation. One of the most interesting generalization, however, has been introduced by F. Riesz in the classical setting, and by Yu. T. Medvedev in the setting of gauge func­ tions because it allows an elegant characterization of absolutely continuous functions whose derivative have certain summability properties. We also consider functions of so-called Korenblum variation, as well as higher order variations in the sense of De la Vallée–Poussin and Popoviciu. All of these generalizations of the concept of bounded variation will be discussed in this chapter, and some relations between them will be illustrated by several inclusions, examples, and counterexamples.

2.1 The Wiener–Young variation In Definition 1.31, we have introduced, for 1 ≤ 𝑝 < ∞, the class 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) of all functions 𝑓 : [𝑎, 𝑏] → ℝ, for which the 𝑝-variation in Wiener’s sense {𝑚 } 𝑝 Var𝑊 𝑝 (𝑓; [𝑎, 𝑏]) = sup { ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| : {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏])} {𝑗=1 } is finite. Now, we consider an important generalization of this concept introduced in 1937 by L. C. Young [322, 323]. To this end, we recall the notion of Young functions (or gauge functions) which we already introduced in Definition 0.16. Definition 2.1. We call a function 𝜙 : [0, ∞) → [0, ∞) Young function (or gauge func­ tion) if 𝜙 is continuous, convex¹, and satisfies² 𝜙(0) = 0, 𝜙(𝑡) > 0 for 𝑡 > 0, and 𝜙(𝑡) → ∞ as 𝑡 → ∞. ◼ Typical examples of Young functions are 𝜙(𝑡) = 𝑡𝑝 for 1 ≤ 𝑝 < ∞, 𝜙(𝑡) = 𝑒𝑡 − 1, or 𝜙(𝑡) = (𝑡 + 1) log(𝑡 + 1).

1 We remark that some authors do not require convexity of a Young function to include such examples like 𝜙(𝑡) = 𝑡𝑝 for 0 < 𝑝 < 1. However, in every important result, convexity of 𝜙 is usually imposed as an additional condition. Sometimes, it is also required that 𝜙 is increasing; however, as we have shown in Lemma 1.36, this is a consequence of the other properties. 2 Here, some authors require that even 𝜙(𝑡)/𝑡 → ∞ as 𝑡 → ∞, but this excludes the example 𝜙(𝑡) = 𝑡. Later (Definition 2.11), we will state this extra condition under the name “condition ∞1 .”

2.1 The Wiener–Young variation

| 113

Definition 2.2. Given a Young function 𝜙 : [0, ∞) → [0, ∞), a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), and a function 𝑓 : [𝑎, 𝑏] → ℝ, the nonnegative real number 𝑚

𝑊 Var𝑊 𝜙 (𝑓, 𝑃) = Var𝜙 (𝑓, 𝑃; [𝑎, 𝑏]) := ∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|)

(2.1)

𝑗=1

is called the Wiener–Young variation of 𝑓 on [𝑎, 𝑏] with respect to 𝑃, while the (possibly infinite) number 𝑊 𝑊 Var𝑊 𝜙 (𝑓) = Var𝜙 (𝑓; [𝑎, 𝑏]) := sup {Var𝜙 (𝑓, 𝑃; [𝑎, 𝑏]) : 𝑃 ∈ P([𝑎, 𝑏])} ,

(2.2)

where the supremum is taken over all partitions of [𝑎, 𝑏], is called the total Wiener– Young variation of 𝑓 on [𝑎, 𝑏]. In case Var𝑊 𝜙 (𝑓; [𝑎, 𝑏]) < ∞, we say that 𝑓 has finite Wiener–Young variation on [𝑎, 𝑏], and write 𝑓 ∈ 𝑉𝜙𝑊 ([𝑎, 𝑏]). ◼ Of course, for 𝜙(𝑡) = 𝑡𝑝 with 1 ≤ 𝑝 < ∞, the set 𝑉𝜙𝑊 ([𝑎, 𝑏]) coincides with the space 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) from Definition 1.31. However, 𝑉𝜙𝑊 ([𝑎, 𝑏]) is in general not a linear space: Example 2.3. The function 𝜙 : [0, ∞) → ℝ defined by 0 { { { −1/𝑡 𝜙(𝑡) := {𝑒 { { 16𝑡−3 { 𝑒4

for 𝑡 = 0 , for 0 < 𝑡 < for 𝑡 ≥

1 4

1 4

,

(2.3)

is a Young function. We claim that the corresponding set 𝑉𝜙𝑊 ([0, 1]) is not a linear space. In fact, consider the function 𝑓 : [0, 1] → ℝ defined by {− 1 𝑓(𝑥) := { 2 log 𝑛 0 {

for 𝑥 =

1 𝑛

(𝑛 = 2, 3, 4, . . .) ,

otherwise .

Observe that the extremal partition for calculating the variation Var𝑊 𝜙 (𝑓; [0, 1]) is 𝑃𝑛 := {0, 𝑠𝑛 , 𝑡𝑛 , 𝑠𝑛−1 , 𝑡𝑛−1 , . . . , 𝑠2 , 𝑡2 , 𝑠1 }

(𝑛 = 1, 2, 3, . . .) ,

where 𝑠𝑘 := 1/𝑘 and 𝑡𝑛 ∈ (𝑠𝑘 , 𝑠𝑘−1 ) is arbitrary. Then we have ∞ ∞ ∞ 1 1 −2 log 𝑘 = 2 ∑ 2 < ∞, Var𝑊 𝜙 (𝑓; [0, 1]) = 2 ∑ 𝜙 [𝑓 ( )] = 2 ∑ 𝑒 𝑘 𝑘 𝑘=2 𝑘=2 𝑘=2

and so 𝑓 ∈ 𝑉𝜙𝑊 ([0, 1]). On the other hand, ∞ ∞ ∞ 1 1 − log 𝑘 Var𝑊 = 2∑ = ∞, 𝜙 (2𝑓; [0, 1]) ≥ 2 ∑ 𝜙 [2𝑓 ( )] = 2 ∑ 𝑒 𝑘 𝑘 𝑘=2 𝑘=2 𝑘=2

and so 2𝑓 ∈ ̸ 𝑉𝜙𝑊 ([0, 1]). This shows that the set 𝑉𝜙𝑊 ([0, 1]) is not a linear space for this choice of 𝜙. ♥

114 | 2 Nonclassical BV-spaces Proposition 2.9 below gives a criterion, both necessary and sufficient, for a Young func­ tion 𝜙 under which the corresponding set 𝑉𝜙𝑊 ([𝑎, 𝑏]) is a linear space. Recall that the appropriate notion for solving this problem for Orlicz classes in Section 0.1 was the 𝛥 2 -condition (for large values of 𝑡, see Definition 0.19 and Proposition 0.20). Here, we need a similar condition for small values of 𝑡: Definition 2.4. A Young function 𝜙 : [0, ∞) → [0, ∞) satisfies a 𝛿2 -condition if 𝜙(2𝑡) ≤ 𝑀𝜙(𝑡)

(0 ≤ 𝑡 ≤ 𝑇)

for suitable constants 𝑀 > 0 and 𝑇 > 0. In this case, we write 𝜙 ∈ 𝛿2 .

(2.4) ◼

Note that the constant 𝑀 appearing in the estimate (2.4) always satisfies 𝑀 ≥ 1. In­ deed, since 𝜙 is increasing, we have 𝜙(𝑡) ≤ 𝜙(2𝑡) ≤ 𝑀𝜙(𝑡) for 0 ≤ 𝑡 ≤ 𝑇, which implies 𝑀 ≥ 1. Now, we state three technical lemmas on Young functions which satisfy a 𝛿2 -con­ dition. The first lemma shows that the bounds 𝑀 and 𝑇 in Definition 2.4 are not inde­ pendent. Lemma 2.5. A function 𝜙 satisfies a 𝛿2 -condition if and only if for each 𝑇󸀠 > 0, there exists a number 𝑀(𝑇󸀠 ) ≥ 1 such that 𝜙(2𝑡) ≤ 𝑀(𝑇󸀠 )𝜙(𝑡)

(0 < 𝑡 ≤ 𝑇󸀠 ) .

Proof. Since the “if” part is trivial, we merely prove the “only if” part. Suppose that 𝜙 satisfies a 𝛿2 -condition. Fix an arbitrary number 𝑇󸀠 such that 𝑇󸀠 ≥ 𝑇/2, where 𝑇 > 0 is as in (2.4). Since 𝜙(𝑇) ≤ 𝑀(𝑇)𝜙(𝑇/2), for 𝑡 ∈ [𝑇/2, 𝑇󸀠 ], we obtain 𝜙(𝑡) ≥

𝜙(𝑇) 1 𝜙(𝑇) 𝜙(𝑡) 𝜙(𝑡) = 𝜙(2𝑡) . 𝑀(𝑇)𝜙(𝑇/2) 𝑀(𝑇) 𝜙(2𝑡) 𝜙(𝑇/2)

(2.5)

On the other hand, keeping the fact in mind that 𝜙 is increasing, from the inequal­ ities 𝑇/2 ≤ 𝑡 ≤ 𝑇󸀠 and 2𝑡 ≤ 2𝑇󸀠 we get 𝜙(2𝑡) 𝜙(𝑡) ≤1≤ . 𝜙(2𝑇󸀠 ) 𝜙(𝑇/2) Consequently, by (2.5), we further obtain 𝜙(𝑡) ≥

𝜙(𝑇) 1 1 𝜙(𝑇) 𝜙(2𝑡) 𝜙(2𝑡) = 𝜙(2𝑡) . 𝑀(𝑇) 𝜙(2𝑡) 𝜙(2𝑇󸀠 ) 𝑀(𝑇) 𝜙(2𝑇󸀠 )

Thus, putting 𝑀(𝑇󸀠 ) := 𝑀(𝑇)

𝜙(2𝑇󸀠 ) 𝜙(𝑇)

and taking into account the estimate 𝜙(2𝑇󸀠 ) ≥ 𝜙(𝑇), we have proved the desired asser­ tion.

2.1 The Wiener–Young variation

| 115

In view of Lemma 2.5, we can assign to every Young function 𝜙 ∈ 𝛿2 the function 𝑀 : (0, ∞) → [1, ∞) defined by 𝑀(𝑇) := sup

0 0)

(2.6)

which is well-defined and increasing on (0, ∞). Lemma 2.6. Let 𝜙 ∈ 𝛿2 . Then 𝜙(𝑠 + 𝑡) ≤ 𝑀(𝑇)[𝜙(𝑠) + 𝜙(𝑡)] for 0 < 𝑠, 𝑡 ≤ 𝑇. Proof. Assuming without loss of generality that 𝑠 < 𝑡, we obtain 𝜙(𝑠 + 𝑡) = 𝜙 (2

𝑠+𝑡 𝑠+𝑡 ) ≤ 𝑀(𝑇)𝜙 ( ) ≤ 𝑀(𝑇)𝜙(𝑡) ≤ 𝑀(𝑇)[𝜙(𝑠) + 𝜙(𝑡)] , 2 2

where we have used (2.6) and the monotonicity of 𝜙. We remark that the more general estimate 𝜙(𝑡1 + 𝑡2 + ⋅ ⋅ ⋅ + 𝑡𝑛 ) ≤ 𝑀𝑛−1 ((𝑛 − 1)𝑇)[𝜙(𝑡1 ) + 𝜙(𝑡2 ) + ⋅ ⋅ ⋅ + 𝜙(𝑡𝑛+1 )] may easily be proved by induction on 𝑛 for 0 < 𝑡𝑖 ≤ 𝑇 (𝑖 = 1, 2, . . . , 𝑛). As a consequence of Lemma 2.6, we derive the following Corollary 2.7. Let 𝜙 ∈ 𝛿2 and let 𝑓, 𝑔 : [𝑎, 𝑏] → ℝ be bounded functions on [𝑎, 𝑏]. Then 𝑊 𝑊 Var𝑊 𝜙 (𝑓 + 𝑔) ≤ 𝑀(2𝐾) [Var𝜙 (𝑓) + Var𝜙 (𝑔)] ,

where 𝐾 := max {‖𝑓‖∞ , ‖𝑔‖∞ }, and ‖ ⋅ ‖∞ denotes the norm (0.39). Moreover, for 𝜇 ∈ ℝ, we have 𝑚 𝑊 Var𝑊 𝜙 (𝜇𝑓) ≤ (𝑚 + 1)𝑀 (2𝑚‖𝑓‖∞ ) Var𝜙 (𝑓) , where 𝑚 = ent |𝜇| = max {𝑘 ∈ ℕ : 𝑘 ≤ |𝜇|} denotes the integer part of |𝜇|. Consequently, 𝜙 ∈ 𝛿2 implies that V𝑊 𝜙 ([𝑎, 𝑏]) is a linear space. 𝑊 Now, we want to compare the function classes V𝑊 𝜙 and V𝜓 for two different Young functions 𝜙 and 𝜓. Let us say that a positive real sequence (𝛽𝑛 )𝑛 is subordinate to an­ other positive real sequence (𝛼𝑛 )𝑛 if the convergence of the series ∑∞ 𝑛=1 𝛼𝑛 implies the convergence of the series ∑∞ 𝛽 . The following lemma is crucial for the comparison 𝑛 𝑛=1 𝑊 of the classes V𝑊 ([𝑎, 𝑏]) and V ([𝑎, 𝑏]). 𝜙 𝜓

Lemma 2.8. Let 𝜙 and 𝜓 be two Young functions. Then, for every sequence (𝑡𝑛 )𝑛 of non­ negative real numbers, the sequence (𝜓(𝑡𝑛 ))𝑛 is subordinate to the sequence (𝜙(𝑡𝑛 ))𝑛 if and only if there exist numbers 𝐴 > 0 and 𝐵 > 0 such that 𝜓(𝑡) ≤ 𝐵𝜙(𝑡)

(0 < 𝑡 ≤ 𝐴) .

(2.7)

116 | 2 Nonclassical BV-spaces Proof. Clearly, (2.7) implies that (𝜓(𝑡𝑛 ))𝑛 is subordinate to (𝜙(𝑡𝑛 ))𝑛 . Conversely, suppose that (2.7) is false. Then for each 𝐴 > 0 and 𝐵 > 0, we can find a number 𝑡 ∈ (0, 𝐴] such that 𝜓(𝑡) > 𝐵𝜙(𝑡). In particular, we may choose 𝐵 = 𝑛 ∈ ℕ and 𝐴 > 0 in such a way that 𝜙(𝐴) = 1/𝑛2 . Denoting by 𝑡𝑛 ∈ (0, 𝐴] the corresponding number satisfying 𝜓(𝑡𝑛 ) > 𝑛𝜙(𝑡𝑛 ), we conclude that ∞

∞

1 < ∞. 2 𝑛=1 𝑛

∑ 𝜙(𝑡𝑛 ) ≤ ∑

𝑛=1

Further, denote by 𝑘𝑛 the least natural number such that 2 1 ≤ 𝑘𝑛 𝜙(𝑡𝑛 ) ≤ 2 . 𝑛2 𝑛 For 𝑚 ∈ ℕ fixed, we choose 𝑛 = 𝑛(𝑚) ∈ ℕ such that 𝑘1 + 𝑘2 + . . . + 𝑘𝑚−1 < 𝑛 ≤ 𝑘1 + 𝑘2 + ⋅ ⋅ ⋅ + 𝑘𝑚 . The sequence (𝑡𝑚 )𝑚 then satisfies ∞

∞

∑ 𝜙(𝑡𝑚 ) < ∞,

∑ 𝜓(𝑡𝑚 ) = ∞ ,

𝑚=1

(2.8)

𝑚=1

contradicting our assumption that (𝜓(𝑡𝑛 ))𝑛 is subordinate to (𝜙(𝑡𝑛 ))𝑛 . We point out that the hypotheses of Lemma 2.8 concerning the numbers 𝐴 and 𝐵 may be stated equivalently in the following way: for each 𝐴󸀠 > 0, we can find a constant 𝐵(𝐴󸀠 ) > 0 such that 𝜓(𝑡) ≤ 𝐵(𝐴󸀠 )𝜙(𝑡) (0 < 𝑡 ≤ 𝐴󸀠 ) ; (2.9) the proof of this fact is similar to that of Lemma 2.5. We are now in a position to prove two important results on the function class 𝑉𝜙𝑊 ([𝑎, 𝑏]). Proposition 2.9. Let 𝜙 and 𝜓 be two Young functions. Then the following hold. (a) The inclusion 𝑉𝜙𝑊 ([𝑎, 𝑏]) ⊆ 𝑉𝜓𝑊 ([𝑎, 𝑏]) holds if and only if (2.7) is true. (b) The set 𝑉𝜙𝑊 ([𝑎, 𝑏]) is a linear space if and only if 𝜙 ∈ 𝛿2 . Proof. (a) Suppose first that (2.7) is true, and so also (2.9), and assume that 𝑓 ∈ 𝑉𝜙𝑊 ([𝑎, 𝑏]). Since 𝑓 is bounded, we may choose 𝐴󸀠 := 2‖𝑓‖∞ in (2.9) and get 𝜓(|𝑓(𝑠) − 𝑓(𝑡)|) ≤ 𝐵(2‖𝑓‖∞ )𝜙(|𝑓(𝑠) − 𝑓(𝑡)|) . 𝑊 𝑊 Consequently, Var𝑊 𝜓 (𝑓) ≤ 𝐵(2‖𝑓‖∞ ) Var𝜙 (𝑓), and so V𝜙 ([𝑎, 𝑏]) ⊆ 𝑉𝜓 ([𝑎, 𝑏]). Conversely, suppose that (2.7) is false, i.e. for every 𝐴 > 0 and 𝐵 > 0, there exists 𝑡 ∈ (0, 𝐴] such that 𝜓(𝑡) > 𝐵𝜙(𝑡). As in Lemma 2.8, we can find a sequence (𝑡𝑚 )𝑚 satisfying (2.8). Denote by (𝑦𝑛 )𝑛 a fixed increasing sequence of points 𝑦𝑛 ∈ (𝑎, 𝑏), and define 𝑓 : [𝑎, 𝑏] → ℝ by

{𝑡𝑚 𝑓(𝑥) := { 0 {

for 𝑥 = 𝑦𝑚 , otherwise .

2.1 The Wiener–Young variation

|

117

We claim that 𝑓 ∈ 𝑉𝜙𝑊 ([𝑎, 𝑏]), but 𝑓 ∈ ̸ 𝑉𝜓𝑊 ([𝑎, 𝑏]). In fact, taking an arbitrary parti­ tion 𝑃 = {𝑥0 , 𝑥1 , . . . , 𝑥𝑚 } ∈ P([𝑎, 𝑏]), we obtain 𝑚

𝑚

𝑚

∞

∑ 𝜙(|𝑓(𝑥𝑗 ) − 𝑓(𝑥𝑗−1 )|) ≤ 2 ∑ 𝜙(𝑡𝑙𝑗 ) + ∑ 𝜙(|𝑡𝑗 − 𝑡𝑗−1 |) ≤ 4 ∑ 𝜙(𝑡𝑗 ) ,

𝑗=1

𝑗=1

𝑗=1

𝑗=1

𝑊 and hence Var𝑊 𝜙 (𝑓) < ∞ and so 𝑓 ∈ 𝑉𝜙 ([𝑎, 𝑏]). On the other hand, consider the parti­ tion 𝑃 = {𝑥0 , 𝑥1 , . . . , 𝑥2𝑚 }, where

𝑥2𝑖+1 = 𝑦𝑖

(𝑖 = 0, 1, 2, . . . , 𝑚 − 2) ,

𝑥2𝑖 =

1 (𝑦 + 𝑦𝑖 ) (𝑖 = 1, 2, . . . , 𝑚 − 1) . 2 𝑖−1

For this partition, we have 2𝑚

𝑚−2

𝑗=1

𝑗=0

Var𝑊 𝜓 (𝑓, 𝑃) = ∑ 𝜓(|𝑓(𝑥𝑗 ) − 𝑓(𝑥𝑗−1 )|) ≥ 2 ∑ 𝜓(𝑡𝑗 ) , 𝑊 and hence Var𝑊 𝜓 (𝑓) = ∞ and so 𝑓 ∈ ̸ 𝑉𝜓 ([𝑎, 𝑏]). (b) The fact that 𝑉𝜙𝑊 ([𝑎, 𝑏]) is a linear space in case 𝜙 ∈ 𝛿2 has already been proved in Corollary 2.7. Conversely, if 𝑉𝜙𝑊 ([𝑎, 𝑏]) is a linear space, then 𝑓 ∈ 𝑉𝜙𝑊 ([𝑎, 𝑏]) implies 2𝑓 ∈ 𝑉𝜙𝑊 ([𝑎, 𝑏]). Thus, putting 𝜓(𝑡) := 𝜙(2𝑡), we conclude that 𝑉𝜙𝑊 ([𝑎, 𝑏]) ⊆ 𝑉𝜓𝑊 ([𝑎, 𝑏]). However, in view of (a), this implies that

𝜙(2𝑡) = 𝜓(𝑡) ≤ 𝐵𝜙(𝑡)

(0 < 𝑡 ≤ 𝐴) ,

and hence 𝜙 ∈ 𝛿2 , and so we are done. Clearly, the function 𝜙(𝑡) = 𝑡𝑝 satisfies 𝜙 ∈ 𝛿2 for 0 < 𝑝 < ∞, as may be seen by choosing 𝑀 = 2𝑝 and 𝑇 = 1 in (2.4). The same is true for 𝜙(𝑡) = log(1 + 𝑡) since lim

𝑡→0+

𝜙(2𝑡) 2(1 + 𝑡) = lim =2 𝑡→0+ 1 + 2𝑡 𝜙(𝑡)

by L’Hospital’s rule. As we have seen in (0.22), the Young function 𝜙(𝑡) = 𝑒𝑡 − 1 does not satisfy a 𝛥 2 -condition. However, it satisfies a 𝛿2 -condition since lim

𝑡→0+

𝜙(2𝑡) = 2 lim 𝑒𝑡 = 2 , 𝑡→0+ 𝜙(𝑡)

again by L’Hospital’s rule. Conversely, the Young function (2.3) satisfies 𝜙 ∈ 𝛥 2 , but 𝜙 ∈ ̸ 𝛿2 because 𝜙(2𝑡) 𝜙(2𝑡) lim = lim 𝑒1/2𝑡 = ∞ , lim = 2. 𝑡→∞ 𝑡→0+ 𝜙(𝑡) 𝑡→0+ 𝜙(𝑡) This explains again why the set 𝑉𝜙𝑊 ([𝑎, 𝑏]) in Example 2.3 is not a linear space. We remark, however, that the set 𝑉𝜙𝑊 ([𝑎, 𝑏]) is always symmetric, balanced, absorbing, and convex. So, we may use the same method as for Orlicz classes and consider the set 𝐵𝑊 (𝜙) := {𝑓 ∈ 𝐵([𝑎, 𝑏]) : Var𝑊 (2.10) 𝜙 (𝑓; [𝑎, 𝑏]) ≤ 1}

118 | 2 Nonclassical BV-spaces together with the corresponding Minkowski functional³ ‖𝑓‖𝑊𝐵𝑉𝜙 := |𝑓(𝑎)| + inf {𝜆 > 0 : 𝑓/𝜆 ∈ 𝐵𝑊 (𝜙)}

(2.11)

which is a norm on⁴ 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) = span 𝑉𝜙𝑊 ([𝑎, 𝑏]). Moreover, the closed unit ball in the space (𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]), ‖ ⋅ ‖𝑊𝐵𝑉𝜙 ) coincides with the set 𝐵𝑊 (𝜙) given in (2.10). Proposition 2.9 (b) shows that, loosely speaking, the 𝛿2 -condition plays, for the space 𝑊𝐵𝑉𝜙 , the same role as the 𝛥 2 -condition for the Orlicz space 𝐿 𝜙 , see Proposi­ tion 0.20. Let us check (2.11) for the function 𝑓 from Example 2.3 in the space 𝑊𝐵𝑉𝜙 ([0, 1]) with 𝜙 as in (2.3). Our calculations in Example 2.3 show that, for 𝜆 > 0, ∞

∞

1 . 2𝜆 𝑘 𝑘=2

−2𝜆 log 𝑘 =2∑ Var𝑊 𝜙 (𝑓/𝜆; [0, 1]) = 2 ∑ 𝑒 𝑘=2

Thus, the infimum⁵ of all 𝜆 > 0 such that Var𝑊 𝜙 (𝑓/𝜆; [0, 1]) is finite is 1/2. To calcu­ late the norm of 𝑓 explicitly, however, we have to find the infimum of all 𝜆 > 0 such that Var𝑊 𝜙 (𝑓/𝜆; [0, 1]) ≤ 1, and this is more difficult. In the following proposition which is parallel to Proposition 1.3, we collect some properties of the quantities (2.1) and (2.2) and the space (𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]), ‖ ⋅ ‖𝑊𝐵𝑉𝜙 ). Proposition 2.10. The quantities (2.1) and (2.2) have the following properties. (a) The variation (2.2) is superadditive with respect to intervals, i.e. 𝑊 𝑊 Var𝑊 𝜙 (𝑓; [𝑎, 𝑏]) ≥ Var𝜙 (𝑓; [𝑎, 𝑐]) + Var𝜙 (𝑓; [𝑐, 𝑏])

for 𝑎 < 𝑐 < 𝑏. (b) Conversely, if 𝜙 satisfies the 𝛿2 -condition (2.4), then 𝑊 𝑊 Var𝑊 𝜙 (𝑓; [𝑎, 𝑏]) ≤ 𝑀(2‖𝑓‖∞ ) [Var𝜙 (𝑓; [𝑎, 𝑐]) + Var𝜙 (𝑓; [𝑐, 𝑏])] ,

for 𝑎 < 𝑐 < 𝑏, where ‖ ⋅ ‖∞ denotes the norm (0.39), and 𝑀 is the function defined in (2.6). (c) If 𝜙 satisfies the 𝛿2 -condition (2.4), then 𝑊 𝑊 Var𝑊 𝜙 (𝑓 + 𝑔; [𝑎, 𝑏]) ≤ 𝑀(2𝐾) [Var𝜙 (𝑓; [𝑎, 𝑏]) + Var𝜙 (𝑔, [𝑎, 𝑏])] ,

where 𝐾 := max {‖𝑓‖∞ , ‖𝑔‖∞ }, and so the variation (2.2) is partly subadditive with respect to functions.

1/𝑝 3 Obviously, for 𝜙(𝑡) = 𝑡𝑝 with 𝑝 ≥ 1 the infimum in (2.11) coincides with Var𝑊 , see (1.61). 𝑝 (𝑓; [𝑎, 𝑏]) So ‖𝑓‖𝑊𝐵𝑉𝜙 = ‖𝑓‖𝑊𝐵𝑉𝑝 in this case, as one should expect.

4 As usual, span 𝑀 denotes the linear hull of 𝑀, i.e. the smallest linear space containing 𝑀. 5 Of course, this infimum is not a minimum! This is precisely the reason for the fact that 2𝑓 ∉ V𝑊 𝜙 ([0, 1]), as we have seen in Example 2.3.

2.1 The Wiener–Young variation

| 119

(d) If 𝜙 satisfies the 𝛿2 -condition (2.4), then 𝑚 𝑊 Var𝑊 𝜙 (𝜇𝑓; [𝑎, 𝑏]) ≤ (𝑚 + 1)𝑀 (2𝑚‖𝑓‖∞ ) Var𝜙 (𝑓; [𝑎, 𝑏])

(𝜇 ∈ ℝ) ,

where 𝑚 = ent |𝜇| denotes the integer part of the number |𝜇|. (e) The estimate Var𝑊 𝜙 (𝑓; [𝑎, 𝑏]) ≤ 𝜙(Var(𝑓; [𝑎, 𝑏])) holds, where Var(𝑓; [𝑎, 𝑏]) denotes the classical variation (1.4). In particular, if 𝑓 is monotone on the interval [𝑎, 𝑏], then Var𝑊 𝜙 (𝑓; [𝑎, 𝑏]) = 𝜙(|𝑓(𝑏) − 𝑓(𝑎)|) . (f) The space 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) is complete with respect to the norm (2.11). Proof. To prove property (a), we may assume that both variations Var𝑊 𝜙 (𝑓; [𝑎, 𝑐]) and 𝑊 Var𝜙 (𝑓; [𝑐, 𝑏]) are finite. Given 𝜀 > 0, choose partitions {𝑡0 , 𝑡1 , . . . , 𝑡𝑘 } ∈ P([𝑎, 𝑐]), such that 𝑡𝑘 = 𝑐,

{𝑡𝑘 , 𝑡𝑘+1 , . . . , 𝑡𝑚 } ∈ P([𝑐, 𝑏]) 𝑘

Var𝑊 𝜙 (𝑓; [𝑎, 𝑐]) − 𝜀 ≤ ∑ 𝜙(|𝑓(𝑡𝑖 ) − 𝑓(𝑡𝑖−1 )|) 𝑖=1

and

𝑚

Var𝑊 𝜙 (𝑓; [𝑐, 𝑏]) − 𝜀 ≤ ∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) . 𝑗=𝑘+1

Then we obtain 𝑊 Var𝑊 𝜙 (𝑓; [𝑎, 𝑐]) + Var𝜙 (𝑓; [𝑐, 𝑏]) − 2𝜀 𝑚

≤ ∑ 𝜙(|𝑓(𝑡𝑘 ) − 𝑓(𝑡𝑘−1 )|) ≤ Var𝑊 𝜙 (𝑓; [𝑎, 𝑏]) 𝑘=1

which proves (a) since 𝜀 > 0 was arbitrary. In order to prove property (b), let {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]). If some point of this partition, say 𝑡𝑘 , coincides with 𝑐, we have 𝑚

𝑘

𝑚

∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) = ∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) + ∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|)

𝑗=1

𝑗=1

≤

Var𝑊 𝜙 (𝑓; [𝑎, 𝑐])

𝑗=𝑘+1

+

Var𝑊 𝜙 (𝑓; [𝑐, 𝑏]) .

Now, taking into account the fact that the function 𝑀 in (2.6) satisfies 𝑀(𝑇) ≥ 1 for all 𝑇 > 0 (Lemma 2.5), we derive the estimate 𝑚

𝑊 ∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) ≤ 𝑀(2‖𝑓‖∞ ) [Var𝑊 𝜙 (𝑓; [𝑎, 𝑐]) + Var𝜙 (𝑓; [𝑐, 𝑏])] .

𝑗=1

(2.12)

120 | 2 Nonclassical BV-spaces Now, consider the opposite case, i.e. 𝑡𝑘−1 < 𝑐 < 𝑡𝑘 for some 𝑘 ∈ {1, 2, . . . , 𝑚}. Then we obtain 𝑚

∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|)

𝑗=1

𝑘−1

𝑚

= ∑ 𝜙 (|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) + 𝜙 (|𝑓(𝑡𝑘 ) − 𝑓(𝑡𝑘−1 )|) + ∑ 𝜙 (|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) 𝑗=1

𝑗=𝑘+1

𝑘−1

= ∑ 𝜙 (|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) + 𝜙 (|𝑓(𝑡𝑘 ) − 𝑓(𝑐)| + |𝑓(𝑐) − 𝑓(𝑡𝑘−1 )|) 𝑗=1

𝑚

+ ∑ 𝜙 (|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) . 𝑗=𝑘+1

From Lemma 2.5 and this estimate, we get 𝑚

∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|)

𝑗=1

𝑘−1

≤ ∑ 𝜙(𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )) + 𝑀(2‖𝑓‖∞ )𝜙(|𝑓(𝑐) − 𝑓(𝑡𝑘−1 )|) 𝑗=1

𝑚

+ 𝑀(2‖𝑓‖∞ )𝜙(|𝑓(𝑡𝑘 ) − 𝑓(𝑐)|) + ∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) , 𝑗=𝑘+1

and hence 𝑚

𝑊 ∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) ≤ 𝑀(2‖𝑓‖∞ ) [Var𝑊 𝜙 (𝑓; [𝑎, 𝑐]) + Var𝜙 (𝑓; [𝑐, 𝑏])] .

(2.13)

𝑗=1

Combining the estimates (2.12) and (2.13), we deduce the desired inequality. The assertions (c) and (d) have already been proved in Corollary 2.7. To prove (e), fix a partition {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]). Then we obtain 𝑚

𝑚

∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) ≤ 𝜙 (∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) ≤ 𝜙(Var(𝑓; [𝑎, 𝑏]))

𝑗=1

𝑗=1

since 𝜙 is superadditive and increasing, see Lemma 1.36. This yields the estimate Var𝑊 𝜙 (𝑓; [𝑎, 𝑏]) ≤ 𝜙(Var(𝑓; [𝑎, 𝑏])) . Obviously, in case of a monotone function 𝑓, we get, in particular, Var(𝑓; [𝑎, 𝑏]) = |𝑓(𝑏) − 𝑓(𝑎)|, so Var𝑊 𝜙 (𝑓; [𝑎, 𝑏]) ≤ 𝜙(|𝑓(𝑏) − 𝑓(𝑎)|) . On the other hand, taking the special partition 𝑃 := {𝑎, 𝑏}, we get 𝑊 𝜙(|𝑓(𝑏) − 𝑓(𝑎)|) = Var𝑊 𝜙 (𝑓, 𝑃; [𝑎, 𝑏]) ≤ Var𝜙 (𝑓; [𝑎, 𝑏])

which proves (e).

2.1 The Wiener–Young variation

| 121

It remains to prove (f). Assume that (𝑓𝑛 )𝑛 is a Cauchy sequence in 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) with respect to the norm (2.11). This means that for 𝜀 > 0, there exists a natural number 𝑛0 such that ‖𝑓𝑛 − 𝑓𝑚 ‖𝑊𝐵𝑉𝜙 ≤ 𝜀 (𝑚, 𝑛 ≥ 𝑛0 ) . Keeping in mind the definition (2.11) of the norm ‖ ⋅ ‖𝑊𝐵𝑉𝜙 , this means that both |𝑓𝑛 (𝑎) − 𝑓𝑚 (𝑎)| ≤ 𝜀 ,

Var𝑊 𝜙 (

𝑓𝑛 − 𝑓𝑚 ; [𝑎, 𝑏]) ≤ 1 . 𝜀

(2.14)

From the first estimate in (2.14), we deduce that the sequence (𝑓𝑛 (𝑎))𝑛 converges to some real number which we denote by 𝑓(𝑎). On the other hand, the second inequality in (2.14) implies that, for any partition {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), we have 𝑚

∑𝜙(

|[𝑓𝑛 (𝑡𝑗 ) − 𝑓𝑚 (𝑡𝑗 )] − [𝑓𝑛 (𝑡𝑗−1 ) − 𝑓𝑚 (𝑡𝑗−1 )]| 𝜀

𝑗=1

) ≤ 1.

(2.15)

For the particular partition {𝑎, 𝑥, 𝑏} (with 𝑥 ∈ (𝑎, 𝑏) fixed), we obtain from (2.15) 𝜙( hence

|[𝑓𝑛 (𝑥) − 𝑓𝑚 (𝑥)] − [𝑓𝑛 (𝑎) − 𝑓𝑚 (𝑎)]| ) ≤ 1, 𝜀

|[𝑓𝑛 (𝑥) − 𝑓𝑚 (𝑥)] − [𝑓𝑛 (𝑎) − 𝑓𝑚 (𝑎)]| ≤ 𝜙−1 (1) , 𝜀

and so |𝑓𝑛 (𝑥) − 𝑓𝑚 (𝑥)| ≤ |𝑓𝑛 (𝑎) − 𝑓𝑚 (𝑎)| + 𝜀𝜙−1 (1) . Finally, for arbitrary 𝑥 ∈ [𝑎, 𝑏], we get |𝑓𝑛 (𝑥) − 𝑓𝑚 (𝑥)| ≤ 𝜀(1 + 𝜙−1 (1)) by the first estimate in (2.14). The last inequality implies that the sequence (𝑓𝑛 )𝑛 is uniformly convergent to some function 𝑓 on the interval [𝑎, 𝑏], i.e. 𝑓𝑛 → 𝑓 in 𝐵([𝑎, 𝑏]). Now, we again take into account the estimate (2.15). Keeping 𝑛 fixed and letting 𝑚 → ∞, we obtain 𝑚

∑𝜙(

|[𝑓𝑛 (𝑡𝑗 ) − 𝑓(𝑡𝑗 )] − [𝑓𝑛 (𝑡𝑗−1 ) − 𝑓(𝑡𝑗−1 )]| 𝜀

𝑗=1

which means nothing else but Var𝑊 𝜙 (

𝑓𝑛 − 𝑓 ; [𝑎, 𝑏]) ≤ 1 𝜀

(𝑛 ≥ 𝑛0 ) ,

and so ‖𝑓𝑛 − 𝑓‖𝑊𝐵𝑉𝜙 ≤ 2𝜀, i.e. 𝑓𝑛 → 𝑓 in 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]).

) ≤ 1,

122 | 2 Nonclassical BV-spaces Note that, in contrast to Proposition 1.3, properties which are analogous to Proposi­ tion 1.3 (f) and (g) are not valid here. Indeed, we have shown this for the special Young function 𝜙(𝑡) = 𝑡𝑝 and the space 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) (𝑝 > 1) already in Example 1.33. Con­ cerning the behavior of Var𝑊 𝑝 (𝑓) with respect to subintervals, a combination of Propo­ sition 2.10 (a) and (b) shows that 𝑊 𝑊 Var𝑊 𝑝 (𝑓; [𝑎, 𝑐]) + Var𝑝 (𝑓; [𝑐, 𝑏]) ≤ Var𝑝 (𝑓; [𝑎, 𝑏]) 𝑊 ≤ 2𝑝 [Var𝑊 𝑝 (𝑓; [𝑎, 𝑐]) + Var𝑝 (𝑓; [𝑐, 𝑏])] ,

where the second estimate follows from the fact that 𝑀(𝑇) ≡ 2𝑝 in (2.6) for 𝜙(𝑡) = 𝑡𝑝 . For further use, we now define an additional property of Young functions in the following Definition 2.11. Let 1 ≤ 𝑝 < ∞. We say that a Young function 𝜙 : [0, ∞) → [0, ∞) satisfies condition ∞𝑝 if 𝜙(𝑡) = ∞. (2.16) lim 𝑡→∞ 𝑡𝑝 In this case, we write 𝜙 ∈ ∞𝑝 . Similarly, we say that 𝜙 satisfies condition 0𝑝 if lim 𝑡→0

𝜙(𝑡) = ∞. 𝑡𝑝

In this case, we write 𝜙 ∈ 0𝑝 .

(2.17) ◼

For example, the Young function 𝜙(𝑡) = 𝑡𝑞 satisfies condition ∞𝑝 for 𝑞 > 𝑝 and condi­ tion 0𝑝 for 𝑞 < 𝑝, while the Young function 𝜙(𝑡) = 𝑒𝑡 − 1 satisfies condition ∞𝑝 for all 𝑝 ≥ 1 and condition 0𝑝 for all 𝑝 > 1. Finally, the Young function 𝜙(𝑡) = (𝑡 + 1) log(𝑡 + 1) satisfies condition ∞𝑝 for 𝑝 = 1, but not for 𝑝 > 1, and condition 0𝑝 for 𝑝 > 1, but not for 𝑝 = 1. In the following Table 2.1, we compare the growth conditions (0.21), (2.4), (2.16) and (2.17) for some Young functions. Table 2.1. Growth properties of Young functions. 𝜙(𝑡) = 𝑝

𝑡 𝑒𝑡−1 (𝑡 + 1) log(𝑡 + 1) (2.3)

𝜙 ∈ 𝛥2

𝜙 ∈ 𝛿2

𝜙 ∈ ∞1

𝜙 ∈ 01

yes no yes yes

yes yes yes no

yes no yes no

no no no yes

Clearly, 𝜙 ∈ ∞𝑝 implies 𝜙 ∈ ∞𝑞 for all 𝑞 < 𝑝, but not vice versa, while 𝜙 ∈ 0𝑝 implies 𝜙 ∈ 0𝑞 for all 𝑞 > 𝑝, but not vice versa. So condition ∞1 is the mildest requirement in (2.16), while condition 01 is the strongest requirement in (2.17). The following proposition gives useful estimates for Young functions 𝜙 ∈ ∞1 .

2.1 The Wiener–Young variation

|

123

Proposition 2.12. Suppose that a Young function 𝜙 : [0, ∞) → [0, ∞) satisfies (2.16) for 𝑝 = 1. Then the following is true. (a) The estimate ∑𝑛 𝛼𝑘 𝑢𝑘 ∑𝑛 𝛼 𝜙(𝑢𝑘 ) 𝜙 ( 𝑘=1 ) ≤ 𝑘=1𝑛 𝑘 (2.18) 𝑛 ∑𝑘=1 𝛼𝑘 ∑𝑘=1 𝛼𝑘 holds for all nonnegative numbers 𝛼1 , . . . , 𝛼𝑛 and 𝑢1 , . . . , 𝑢𝑛 satisfying 𝑛

∑ 𝛼𝑘 > 0 .

𝑘=1

(b) The estimate

𝑑

𝜙(

∫𝑐 𝛼(𝑡)𝑢(𝑡) 𝑑𝑡 𝑑

𝑑

)≤

∫𝑐 𝛼(𝑡)𝜙(𝑢(𝑡)) 𝑑𝑡

∫𝑐 𝛼(𝑡) 𝑑𝑡

𝑑

(2.19)

∫𝑐 𝛼(𝑡) 𝑑𝑡

holds for all nonnegative integrable functions 𝑢 : [𝑐, 𝑑] → ℝ and 𝛼 : [𝑐, 𝑑] → ℝ satisfying 𝑑

∫ 𝛼(𝑡) 𝑑𝑡 > 0 . 𝑐

Proof. For 𝑛 = 2, the estimate (2.18) is just the definition of the convexity of 𝜙; for 𝑛 ≥ 3, we may prove assertion (a) easily by induction. To prove (b), let 𝑣 > 0 be fixed. From the convexity of 𝜙, it follows that there exists a 𝑘 ∈ ℝ such that 𝜙(𝑢) − 𝜙(𝑣) ≥ 𝑘(𝑢 − 𝑣) (𝑢 ≥ 0) . Substituting 𝑢 = 𝑢(𝑡) but keeping 𝑣 constant, multiplying by 𝛼(𝑡) and integrating over [𝑐, 𝑑] yields 𝑑

𝑑

𝑑

𝑑

{ } ∫ 𝛼(𝑡)𝜙(𝑢(𝑡)) 𝑑𝑡 − 𝜙(𝑣) ∫ 𝛼(𝑡) 𝑑𝑡 ≥ 𝑘 {∫ 𝛼(𝑡)𝑢(𝑡) 𝑑𝑡 − 𝑣 ∫ 𝛼(𝑡) 𝑑𝑡} . 𝑐 𝑐 𝑐 {𝑐 } Now, for the choice

(2.20)

𝑑

𝑣 :=

∫𝑐 𝛼(𝑡)𝑢(𝑡) 𝑑𝑡 𝑑

,

∫𝑐 𝛼(𝑡) 𝑑𝑡 (2.20) becomes 𝑑

∫ 𝛼(𝑡)𝜙(𝑢(𝑡)) 𝑑𝑡 − 𝜙 ( 𝑐

𝑑

∫𝑐 𝛼(𝑡)𝑢(𝑡) 𝑑𝑡 𝑑

∫𝑐 𝛼(𝑡) 𝑑𝑡

𝑑

) ∫ 𝛼(𝑡) 𝑑𝑡 ≥ 0 , 𝑐

and the estimate (2.19) follows. The estimate (2.18) is usually called the discrete Jensen inequality and the estimate (2.19) is the continuous Jensen inequality.

124 | 2 Nonclassical BV-spaces 1/𝑝 As the definition (1.65) of ‖ ⋅ ‖𝑊𝐵𝑉𝑝 shows, the expression Var𝑊 plays 𝑝 (𝑓; [𝑎, 𝑏]) an important role since it is homogeneous in 𝑓, see Proposition 1.32 (b). Now, we pro­ vide a result concerning the relationship between two spaces 𝑊𝐵𝑉𝜙 and 𝑊𝐵𝑉𝜓 which involves a similar expression for general Young functions 𝜙 and 𝜓. This result gener­ alizes Proposition 1.38.

Proposition 2.13. Let 𝜙 and 𝜓 be Young functions such that the function 𝜓 ∘ 𝜙−1 is con­ vex. Then the inequality −1 𝑊 𝜓−1 (Var𝑊 𝜓 (𝑓; [𝑎, 𝑏])) ≤ 𝜙 (Var𝜙 (𝑓; [𝑎, 𝑏]))

(2.21)

𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) ⊆ 𝑊𝐵𝑉𝜓 ([𝑎, 𝑏])

(2.22)

holds. Consequently, for such Young functions 𝜙 and 𝜓. Proof. To prove (2.21), we may assume that Var𝑊 𝜙 (𝑓; [𝑎, 𝑏]) is finite. Fix an arbitrary partition {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]). Then, keeping in mind the fact that both functions 𝜓 and 𝜙−1 are increasing, see Lemma 1.36, and taking into account (1.70) (with 𝜙 replaced by 𝜓 ∘ 𝜙−1 ), we obtain 𝑚

Var𝑊 𝜓 (𝑓, 𝑃; [𝑎, 𝑏]) = ∑ 𝜓(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) 𝑗=1 𝑚

𝑚

= ∑ (𝜓 ∘ 𝜙−1 ∘ 𝜙)(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) = ∑ (𝜓 ∘ 𝜙−1 )(𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|)) 𝑗=1

𝑗=1

𝑚

≤ (𝜓 ∘ 𝜙−1 ) (∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|)) 𝑗=1

−1

−1 𝑊 = (𝜓 ∘ 𝜙 ) (Var𝑊 𝜙 (𝑓, 𝑃; [𝑎, 𝑏])) ≤ (𝜓 ∘ 𝜙 ) (Var𝜙 (𝑓; [𝑎, 𝑏])) .

Passing in the first expression to the supremum over all partitions, we get −1 𝑊 Var𝑊 𝜓 (𝑓; [𝑎, 𝑏]) ≤ 𝜓 (𝜙 (Var𝜙 (𝑓; [𝑎, 𝑏]))) ,

and applying the monotone function 𝜓−1 to both sides gives (2.21). The inclusion (2.22) is of course an immediate consequence of (2.21). Putting 𝜙(𝑡) = 𝑡 in (2.21), the convexity of 𝜓 = 𝜓 ∘ 𝜙−1 implies the following Corollary 2.14. Let 𝜓 be a Young function. Then the inequality 𝜓−1 (Var𝑊 𝜓 (𝑓; [𝑎, 𝑏])) ≤ Var(𝑓; [𝑎, 𝑏]) holds for arbitrary bounded functions 𝑓 : [𝑎, 𝑏] → ℝ. Consequently, the inclusion 𝐵𝑉([𝑎, 𝑏]) ⊆ 𝑊𝐵𝑉𝜓 ([𝑎, 𝑏]) holds true.

(2.23)

2.2 The Waterman variation

|

125

Observe that (2.23) means that every function of classical bounded variation on [𝑎, 𝑏] has a bounded Wiener–Young variation with respect to an arbitrary Young function 𝜙. Of course, the simplest examples for Young functions satisfying the hypotheses of Proposition 2.13 are 𝜙(𝑡) = 𝑡𝑝 and 𝜓(𝑡) = 𝑡𝑞 with 1 ≤ 𝑝 ≤ 𝑞. In this way, we get Proposi­ tion 1.38 as a special case of Proposition 2.13.

2.2 The Waterman variation In Section 1.2, we have introduced the family 𝛴([𝑎, 𝑏]) of all finite collections 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} of nonoverlapping intervals [𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ] ⊂ [𝑎, 𝑏], and the family 𝛴∞ ([𝑎, 𝑏]) of all infinite collections 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} of nonoverlapping intervals [𝑎𝑛 , 𝑏𝑛 ] ⊂ [𝑎, 𝑏]. These families still may be used to introduce another concept of variation. Definition 2.15. A Waterman sequence is a decreasing sequence Λ = (𝜆 𝑛 )𝑛 of positive real numbers such that 𝜆 𝑛 → 0 as 𝑛 → ∞ and⁶ ∞

(2.24)

∑ 𝜆𝑛 = ∞ .

𝑛=1

Given a function 𝑓 : [𝑎, 𝑏] → ℝ, a set 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]), and a Waterman sequence Λ = (𝜆 𝑛 )𝑛 , the positive real number ∞

VarΛ (𝑓, 𝑆∞ ) = VarΛ (𝑓, 𝑆∞ ; [𝑎, 𝑏]) := ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|

(2.25)

𝑘=1

is called the Waterman variation of 𝑓 on [𝑎, 𝑏] with respect to 𝑆∞ , while the (possibly infinite) number VarΛ (𝑓) = VarΛ (𝑓; [𝑎, 𝑏]) := sup {VarΛ (𝑓, 𝑆∞ ; [𝑎, 𝑏]) : 𝑆∞ ∈ 𝛴∞ ([𝑎, 𝑏])} ,

(2.26)

where the supremum is taken over all collections 𝑆∞ ∈ 𝛴∞ ([𝑎, 𝑏]), is called the to­ tal Waterman variation of 𝑓 on [𝑎, 𝑏]. In case VarΛ (𝑓; [𝑎, 𝑏]) < ∞, we say that 𝑓 has bounded Waterman variation (or bounded Λ-variation in Waterman’s sense) on [𝑎, 𝑏] and write 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]). ◼ If we drop the condition 𝜆 𝑛 → 0 and take 𝜆 𝑛 ≡ 1, the space Λ𝐵𝑉 coincides with the classical space 𝐵𝑉. Proposition 2.17 (e) below shows that this is, in a certain sense, an “extremal” choice for Λ. We also remark that Λ𝐵𝑉 and 𝐵𝑉 coincide if and only if the sequence (𝜆 𝑛 )𝑛 is bounded away from zero (Exercise 2.4). This is the reason why we require 𝜆 𝑛 → 0 as 𝑛 → ∞ in Definition 2.15. 6 The most natural example of such a sequence is of course 𝜆 𝑛 = 𝑛−𝑞 for 0 < 𝑞 ≤ 1; we will consider this case separately in Definition 2.29 below.

126 | 2 Nonclassical BV-spaces Before describing some properties of the Waterman variation (2.26), we prove an auxiliary fact which is parallel to Proposition 1.18 and will be used later. Proposition 2.16 (localization principle). Suppose that 𝑓 ∈ ̸ Λ𝐵𝑉([𝑎, 𝑏]). Then there ex­ ists a point 𝑥0 ∈ [𝑎, 𝑏] such that 𝑓 ∈ ̸ Λ𝐵𝑉([𝑐, 𝑑]) for each interval [𝑐, 𝑑] ⊆ [𝑎, 𝑏] such that 𝑥0 ∈ (𝑐, 𝑑). Proof. We successively divide the interval 𝐼 := [𝑎, 𝑏] into two halves. Suppose that the series in (2.25) is uniformly bounded by some constant 𝑀 > 0 whenever the intervals 𝐼𝑛 := [𝑎𝑛 , 𝑏𝑛 ] are all contained either in the lower half of 𝐼 or all in the upper half of 𝐼. Then VarΛ (𝑓; [𝑎, 𝑏]) ≤ 2𝑀, contradicting our assumption. Therefore, the set of such series must be unbounded in one half of the interval 𝐼. If we continue this procedure we obtain a nested sequence of intervals (𝐽𝑛 )𝑛 converging to some point 𝑥0 . Now, if 𝑥0 is an interior point of some subinterval 𝐽 ⊆ 𝐼, then 𝐽 ⊇ 𝐽𝑛 for 𝑛 sufficiently large. From this, the conclusion follows. Now, we collect some simple properties of the Waterman variations (2.25) and (2.26) for further reference. Proposition 2.17. The quantities (2.25) and (2.26) have the following properties. (a) The Waterman variation (2.25) is monotone with respect to collections of subinter­ vals, i.e. 𝑆∞ ⊆ 𝑇∞ implies VarΛ (𝑓, 𝑆∞ ) ≤ VarΛ (𝑓, 𝑇∞ ) . (b) The total Waterman variation (2.26) is subadditive with respect to intervals, i.e. VarΛ (𝑓; [𝑎, 𝑏]) ≤ VarΛ (𝑓; [𝑎, 𝑐]) + VarΛ (𝑓; [𝑐, 𝑏]) . (c) The total Waterman variation (2.26) is positively homogeneous, i.e. the equality VarΛ (𝜇𝑓; [𝑎, 𝑏]) = |𝜇| VarΛ (𝑓; [𝑎, 𝑏]) holds for all 𝜇 ∈ ℝ. (d) The variation (2.26) is subadditive with respect to functions, i.e. VarΛ (𝑓 + 𝑔; [𝑎, 𝑏]) ≤ VarΛ (𝑓; [𝑎, 𝑏]) + VarΛ (𝑔; [𝑎, 𝑏]) ; moreover, | VarΛ (𝑓; [𝑎, 𝑏]) − VarΛ (𝑔; [𝑎, 𝑏])| ≤ VarΛ (𝑓 − 𝑔; [𝑎, 𝑏]) . (e) If 𝑓 is of bounded variation on the interval [𝑎, 𝑏], then 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]) for every Waterman sequence Λ. (f) If 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]), then 𝑓 is bounded on [𝑎, 𝑏]. Proof. Property (a) is obvious. To prove (b), fix an arbitrary collection 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]). We define two index sets 𝑁1 , 𝑁2 ⊂ ℕ by 𝑁1 := {𝑘 ∈ ℕ : [𝑎𝑘 , 𝑏𝑘 ] ⊆ [𝑎, 𝑐]},

𝑁2 := {𝑘 ∈ ℕ : [𝑎𝑘 , 𝑏𝑘 ] ⊆ [𝑐, 𝑏]} .

2.2 The Waterman variation

| 127

Obviously, the sets 𝑁1 and 𝑁2 are disjoint. Moreover, there exists at most one nat­ ural number 𝑘0 such that 𝑐 ∈ (𝑎𝑘0 , 𝑏𝑘0 ). So, we have ∞

∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| =

𝑘=1

∑ 𝑘∈𝑁1 ∪𝑁2 ∪{𝑘0 }

𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|

= ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| 𝑘∈𝑁1

+ ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| + 𝜆 𝑘0 |𝑓(𝑏𝑘0 ) − 𝑓(𝑎𝑘0 )| 𝑘∈𝑁2

{ } ≤ { ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| + 𝜆 𝑘0 |𝑓(𝑏𝑘0 ) − 𝑓(𝑐)|} {𝑘∈𝑁1 } { } + { ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| + 𝜆 𝑘0 |𝑓(𝑐) − 𝑓(𝑎𝑘0 )|} {𝑘∈𝑁2 } ≤ VarΛ (𝑓; [𝑎, 𝑐]) + VarΛ (𝑓; [𝑐, 𝑏]) . This yields the desired inequality from property (b). The proof of (c) is trivial and therefore omitted. To prove (d), fix an infinite collection 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]). Then ∞

∑ 𝜆 𝑛 |𝑓(𝑏𝑛 ) + 𝑔(𝑏𝑛 ) − 𝑓(𝑎𝑛 ) − 𝑔(𝑎𝑛 )|

𝑛=1

∞

∞

≤ ∑ 𝜆 𝑛 |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| + ∑ 𝜆 𝑛|𝑔(𝑏𝑛 ) − 𝑔(𝑎𝑛 )| , 𝑛=1

𝑛=1

and hence VarΛ (𝑓 + 𝑔) ≤ VarΛ (𝑓) + VarΛ (𝑔) . In the case when 𝑓, 𝑔 ∈ Λ𝐵𝑉([𝑎, 𝑏]), from this inequality, we deduce that VarΛ (𝑓) = VarΛ [(𝑓 − 𝑔) + 𝑔] ≤ VarΛ (𝑓 − 𝑔) + VarΛ (𝑔) , and hence VarΛ (𝑓) − VarΛ (𝑔) ≤ VarΛ (𝑓 − 𝑔) . In the same way, keeping in mind property (c) for 𝜇 = −1, we get VarΛ (𝑔) − VarΛ (𝑓) ≤ VarΛ (𝑓 − 𝑔) , and therefore (d) is proved. For the proof of (e), we again fix an infinite collection 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]). Since the sequence Λ = (𝜆 𝑛 )𝑛 is decreasing, we obtain ∞

∞

∑ 𝜆 𝑛|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| ≤ 𝜆 1 ∑ |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| ≤ 𝜆 1 Var(𝑓; [𝑎, 𝑏]) .

𝑛=1

𝑛=1

128 | 2 Nonclassical BV-spaces From this property (e) follows. Finally, to prove (f), suppose that 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]), but 𝑓 is not bounded on the interval [𝑎, 𝑏]. Then there exists a sequence (𝑢𝑛 )𝑛 of points 𝑢𝑛 ∈ [𝑎, 𝑏] such that |𝑓(𝑢𝑛 )| → ∞. Since (𝑢𝑛 )𝑛 is bounded, it is possible to select a subsequence (𝑣𝑛 )𝑛 of the sequence (𝑢𝑛 )𝑛 such that 𝑣𝑛 → 𝑥 for some 𝑥 ∈ [𝑎, 𝑏], but still |𝑓(𝑣𝑛 )| → ∞ as 𝑛 → ∞. Next, we can choose a monotone subsequence (𝑤𝑛 )𝑛 of (𝑣𝑛 )𝑛 with |𝑓(𝑤𝑛 )| → ∞ and 𝑤𝑛 → 𝑥, and then another subsequence (𝑧𝑛 )𝑛 of (𝑤𝑛 )𝑛 such that |𝑓(𝑧𝑛+1 )| ≥ 1 + |𝑓(𝑧𝑛 )|

(𝑛 = 1, 2, 3, . . .) .

(2.27)

Now, the intervals [𝑧1 , 𝑧2 ], [𝑧2 , 𝑧3 ], [𝑧3 , 𝑧4 ], . . . form a collection 𝑆∞ ∈ 𝛴∞ ([𝑎, 𝑏]). Calculating the Waterman variation (2.25) with respect to this collection, we get ∞

VarΛ (𝑓, 𝑆∞ ) = ∑ 𝜆 𝑛|𝑓(𝑧𝑛+1 ) − 𝑓(𝑧𝑛 )| 𝑛=1 ∞

∞

≥ ∑ 𝜆 𝑛 [|𝑓(𝑧𝑛+1 )| − |𝑓(𝑧𝑛 )|] ≥ ∑ 𝜆 𝑛 = ∞ 𝑛=1

𝑛=1

by (2.27). However, this contradicts the assumption that 𝑓 belongs to Λ𝐵𝑉([𝑎, 𝑏]), and the proof is complete. Proposition 2.17 (f) may be formulated as imbedding Λ𝐵𝑉 󳨅→ 𝐵, where the imbedding constant 𝑐(Λ𝐵𝑉, 𝐵) only involves the first element 𝜆 1 of Λ, see Exercise 2.17. Note that properties (e) and (f) in Proposition 2.17 may be summarized as inclusions 𝐵𝑉([𝑎, 𝑏]) ⊆ ⋂ Λ𝐵𝑉([𝑎, 𝑏]), Λ

⋃ Λ𝐵𝑉([𝑎, 𝑏]) ⊆ 𝐵([𝑎, 𝑏]) ,

(2.28)

Λ

where the intersection and union in (2.28) are taken over all Waterman sequences Λ. Later (Proposition 2.24), we will prove a more precise result. In the following proposition, we state yet another useful characterization of func­ tions from Λ𝐵𝑉([𝑎, 𝑏]). Proposition 2.18. The following three statements are equivalent. (a) The function 𝑓 belongs to Λ𝐵𝑉([𝑎, 𝑏]). (b) There exists a constant 𝑀 > 0 such that ∞

VarΛ (𝑓, 𝑆∞ ) = ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| ≤ 𝑀 𝑘=1

for every infinite collection 𝑆∞ = {[𝑎𝑘 , 𝑏𝑘 ] : 𝑘 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]). (c) There exists a constant 𝑁 > 0 such that 𝑛

VarΛ (𝑓, 𝑆) = ∑ 𝜆 𝑖 |𝑓(𝑏𝑖 ) − 𝑓(𝑎𝑖 )| ≤ 𝑁 𝑖=1

for any finite collection 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏]).

2.2 The Waterman variation

|

129

Proof. Clearly, (a) is equivalent to (b) in view of Definition 2.15. The fact that (c) im­ plies (b) is a simple consequence of a necessary, sufficient and well-known conver­ gence condition for series with nonnegative terms. So, we only have to prove that (b) implies (c). Assume that condition (b) is satisfied, but (c) is not. This means that for any num­ ber 𝑁 > 0, there exists a finite collection 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏]) such that 𝑛

∑ 𝜆 𝑖 |𝑓(𝑏𝑖 ) − 𝑓(𝑎𝑖 )| > 𝑁 . 𝑖=1

Taking 𝑁 := 2𝑀, where 𝑀 is the constant appearing in (b), we find a finite collec­ tion 𝑆𝑀 := {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛]} ∈ 𝛴([𝑎, 𝑏]) satisfying 𝑛

Var(𝑓, 𝑆𝑀 ) = ∑ 𝜆 𝑖 |𝑓(𝑏𝑖 ) − 𝑓(𝑎𝑖 )| > 2𝑀 . 𝑖=1

Now, we distinguish two possible cases. Assume first that [𝑎1 , 𝑏1 ] ∪ [𝑎2 , 𝑏2 ] ∪ . . . ∪ [𝑎𝑛 , 𝑏𝑛 ] = [𝑎, 𝑏] . Consider an infinite collection 𝑆∗∞ = {[𝛼𝑖 , 𝛽𝑖 ] : 𝑖 = 𝑛, 𝑛 + 1, 𝑛 + 2, . . .} ∈ 𝛴∞ ([𝑎𝑛 , 𝑏𝑛 ]) whose first element satisfies 𝛼𝑛 = 𝑎𝑛 ,

𝛽𝑛 =

𝑎𝑛 + 𝑏𝑛 . 2

Adding to this collection the reduced collection 𝑆󸀠𝑀 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛−1 , 𝑏𝑛−1 ]}, we get the infinite collection 𝑆󸀠𝑀 ∪ 𝑆∗∞ = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛−1 , 𝑏𝑛−1 ], [𝛼𝑛 , 𝛽𝑛 ], [𝛼𝑛+1 , 𝛽𝑛+1 ], . . .} ∈ 𝛴∞ ([𝑎, 𝑏]) . By assumption (b), we know that 𝑛−1

∞

𝑘=1

𝑘=𝑛

VarΛ (𝑓, 𝑆󸀠𝑀 ∪ 𝑆∗∞ ) = ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| + ∑ 𝜆 𝑘 |𝑓(𝛽𝑘 ) − 𝑓(𝛼𝑘 )| ≤ 𝑀 . By only taking into account the first term in the last sum, we derive the estimate 𝑛−1

∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| + 𝜆 𝑛 |𝑓((𝑎𝑛 + 𝑏𝑛 )/2) − 𝑓(𝑎𝑛 )| ≤ 𝑀 .

𝑘=1

Similarly, we now consider an infinite collection 𝑆∗∗ ∞ = {[𝛾𝑖 , 𝛿𝑖 ] : 𝑖 = 𝑛, 𝑛 + 1, 𝑛 + 2, . . .} ∈ 𝛴∞ ([𝑎𝑛 , 𝑏𝑛 ]) whose first element satisfies 𝛾𝑛 =

𝑎𝑛 + 𝑏𝑛 , 2

𝛿𝑛 = 𝑏𝑛 .

Adding to this collection again the reduced collection 𝑆󸀠𝑀 , we get the infinite col­ lection 𝑆󸀠𝑀 ∪ 𝑆∗∗ ∞ = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛−1 , 𝑏𝑛−1 ], [𝛾𝑛 , 𝛿𝑛 ], [𝛾𝑛+1 , 𝛿𝑛+1 ], . . .} ∈ 𝛴∞ ([𝑎, 𝑏]) .

130 | 2 Nonclassical BV-spaces Using the same argument as before, we then obtain the estimate 𝑛−1

∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| + 𝜆 𝑛|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 + 𝑏𝑛 )/2)| ≤ 𝑀 . 𝑘=1

Combining both estimates obtained in this way and putting 𝑐𝑛 := (𝑎𝑛 + 𝑏𝑛 )/2, we end up with 𝑛−1

2 ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| + 𝜆 𝑛 |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| 𝑘=1 𝑛−1

≤ 2 ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| + 𝜆 𝑛 |𝑓(𝑏𝑛 ) − 𝑓(𝑐𝑛 )| + 𝜆 𝑛|𝑓(𝑐𝑛 ) − 𝑓(𝑎𝑛 )| 𝑘=1

𝑛−1

= ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| + 𝜆 𝑛 |𝑓(𝑏𝑛 ) − 𝑓(𝑐𝑛 )| 𝑘=1 𝑛−1

+ ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| + 𝜆 𝑛 |𝑓(𝑐𝑛 ) − 𝑓(𝑎𝑛 )| ≤ 2𝑀 , 𝑘=1

contradicting our choice of 𝑆𝑀 . Now, assume that we have the strict inclusion [𝑎1 , 𝑏1 ] ∪ [𝑎2 , 𝑏2 ] ∪ . . . ∪ [𝑎𝑛 , 𝑏𝑛 ] ⊂ [𝑎, 𝑏] . Choose an interval [𝛼, 𝛽] ⊂ [𝑎, 𝑏] \ ([𝑎1 , 𝑏1 ] ∪ [𝑎2 , 𝑏2 ] ∪ . . . ∪ [𝑎𝑛 , 𝑏𝑛 ]) and a sequence [𝑎𝑛+1 , 𝑏𝑛+1 ], [𝑎𝑛+2 , 𝑏𝑛+2 ], . . . of nonoverlapping intervals such that 𝑆∞ := {[𝑎𝑗 , 𝑏𝑗 ] : 𝑗 = 𝑛 + 1, 𝑛 + 2, . . .} ∈ 𝛴∞ ([𝑎, 𝑏]). Then for the union 𝑆𝑀 ∪ 𝑆∞ = {[𝑎𝑘 , 𝑏𝑘 ] : 𝑘 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]) of the collections 𝑆𝑀 and 𝑆∞ , we obtain ∞

VarΛ (𝑓, 𝑆𝑀 ∪ 𝑆∞ ) = ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| 𝑘=1 𝑛

∞

= ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| + ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| ≤ 𝑀 . 𝑘=1

𝑘=𝑛+1

However, this implies 𝑛

VarΛ (𝑓, 𝑆𝑀 ) = ∑ 𝜆 𝑖 |𝑓(𝑏𝑖 ) − 𝑓(𝑎𝑖 )| ≤ 𝑀 , 𝑖=1

again contradicting our choice 𝑆𝑀 . Thus, in both cases, the proof is complete.

2.2 The Waterman variation

| 131

Proposition 2.18 shows that (2.26) could be equivalently replaced by VarΛ (𝑓) = VarΛ (𝑓; [𝑎, 𝑏]) := sup {VarΛ (𝑓, 𝑆; [𝑎, 𝑏]) : 𝑆 ∈ 𝛴([𝑎, 𝑏])}

(2.29)

in the definition of the Waterman space Λ𝐵𝑉([𝑎, 𝑏]). Proposition 2.19. The set Λ𝐵𝑉([𝑎, 𝑏]) is a linear space; equipped with the norm ‖𝑓‖Λ𝐵𝑉 := |𝑓(𝑎)| + VarΛ (𝑓; [𝑎, 𝑏]),

(2.30)

it is a Banach space. Proof. The fact that Λ𝐵𝑉([𝑎, 𝑏]) is a linear space follows from Proposition 2.17 (c) and (d). We show that the space (Λ𝐵𝑉([𝑎, 𝑏]), ‖ ⋅ ‖Λ𝐵𝑉 ) is complete. To this end, assume that (𝑓𝑛 )𝑛 is a Cauchy sequence in the norm (2.30). Given 𝜀 > 0, choose 𝑛0 ∈ ℕ such that |𝑓𝑛 (𝑎) − 𝑓𝑚 (𝑎)| ≤ 𝜀

(2.31)

VarΛ (𝑓𝑛 − 𝑓𝑚 ) ≤ 𝜀

(2.32)

and for 𝑚, 𝑛 ≥ 𝑛0 . From (2.31), we deduce that the sequence (𝑓𝑛 (𝑎))𝑛 converges to some real number, say 𝑓(𝑎). On the other hand, (2.32) implies that for any infinite collection 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]), we have ∞

VarΛ (𝑓𝑛 − 𝑓𝑚 , 𝑆∞ ) = ∑ 𝜆 𝑘 |𝑓𝑛 (𝑏𝑘 ) − 𝑓𝑚 (𝑏𝑘 ) − 𝑓𝑛 (𝑎𝑘 ) + 𝑓𝑚 (𝑎𝑘 )| ≤ 𝜀 . 𝑘=1

In particular, for any 𝑥 ∈ [𝑎, 𝑏], the inequality 𝜆 1 |𝑓𝑛 (𝑥) − 𝑓𝑚 (𝑥)| − 𝜆 1 |𝑓𝑛 (𝑎) − 𝑓𝑚 (𝑎)| ≤ 𝜆 1 |[𝑓𝑛 (𝑥) − 𝑓𝑚 (𝑥)] − [𝑓𝑛 (𝑎) − 𝑓𝑚 (𝑎)]| ≤ 𝜀 is satisfied. Combining this estimate with (2.31), we obtain |𝑓𝑛 (𝑥) − 𝑓𝑚 (𝑥)| ≤ 𝜀

1 + 𝜆1 , 𝜆1

which shows that the sequence (𝑓𝑛 )𝑛 converges uniformly to some bounded function 𝑓 on the interval [𝑎, 𝑏]. Thus, from the inequality | VarΛ (𝑓𝑛 ) − VarΛ (𝑓𝑚 )| ≤ VarΛ (𝑓𝑛 − 𝑓𝑚 ) , (Proposition 2.17 (d), and (2.32)) it follows that the sequence (VarΛ (𝑓𝑛 ))𝑛 has a limit in ℝ. Now, we fix an arbitrary finite collection 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑝 , 𝑏𝑝 ]} ∈ 𝛴([𝑎, 𝑏]). From the definition of the function 𝑓, we see that, for 𝑛 sufficiently large, we have 𝑝

𝑝

∑ 𝜆 𝑖 |𝑓(𝑏𝑖 ) − 𝑓(𝑎𝑖 )| ≤ ∑ 𝜆 𝑖 |𝑓𝑛 (𝑏𝑖 ) − 𝑓𝑛 (𝑎𝑖 )| + 𝜀 ≤ VarΛ (𝑓𝑛 ) + 𝜀 . 𝑖=1

𝑖=1

132 | 2 Nonclassical BV-spaces Therefore, letting 𝑛 → ∞, we obtain VarΛ (𝑓) ≤ lim VarΛ (𝑓𝑛 ) , 𝑛→∞

and so 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]). Using again the Cauchy property (2.32) of the sequence (𝑓𝑛 )𝑛 , for 𝜀 > 0, we choose 𝑛0 ∈ ℕ such that given any infinite collection 𝑆∞ = {[𝑎𝑘 , 𝑏𝑘 ] : 𝑘 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]), we have ∞

VarΛ (𝑓, 𝑆∞ ) = ∑ 𝜆 𝑘 |𝑓𝑛 (𝑏𝑘 ) − 𝑓𝑚 (𝑏𝑘 ) − 𝑓𝑛 (𝑎𝑘 ) + 𝑓𝑚 (𝑎𝑘 )| ≤ 𝜀 . 𝑘=1

Letting 𝑚 → ∞ and taking the supremum over all such collections 𝑆∞ yields VarΛ (𝑓𝑛 − 𝑓) ≤ 𝜀 . Obviously, letting in (2.31) 𝑚 → ∞, we also get |𝑓𝑛 (𝑎) − 𝑓(𝑎)| ≤ 𝜀, and so ‖𝑓 − 𝑓𝑛 ‖Λ𝐵𝑉 → 0 as 𝑛 → ∞, and the proof is complete. As earlier announced, the relation (2.28) between the spaces 𝐵𝑉 and Λ𝐵𝑉 can be made more precise. In fact, we are now going to prove the equalities ⋂ Λ𝐵𝑉([𝑎, 𝑏]) = 𝐵𝑉([𝑎, 𝑏]), Λ

⋃ Λ𝐵𝑉([𝑎, 𝑏]) = 𝑅([𝑎, 𝑏]) ,

(2.33)

Λ

where 𝑅([𝑎, 𝑏]) denotes the space of all regular functions, see Section 0.3, and the in­ tersection and union in (2.33) are taken over all Waterman sequences Λ. First, we need some auxiliary facts which we collect in a series of technical lemmas. Lemma 2.20. Let (𝛼𝑛 )𝑛 be a sequence of positive real numbers tending to zero. Then there exists a Waterman sequence Λ = (𝜆 𝑛 )𝑛 such that ∞

(2.34)

∑ 𝛼𝑛 𝜆 𝑛 < ∞ .

𝑛=1

Proof. Denote 𝑛0 := 0, choose a natural number 𝑛1 such that 𝛼𝑛 ≤ 1/2 for 𝑛 ≥ 𝑛1 , and put 𝜆 𝑛 := 1/𝑛1 for 𝑛0 < 𝑛 ≤ 𝑛1 . Assume that we have chosen natural numbers 𝑛1 , 𝑛2 , . . . , 𝑛𝑘 in such a way that 𝑛1 < 𝑛2 < . . . < 𝑛𝑘 , 𝛼𝑛 ≤ 1/2𝑘 for 𝑛 ≥ 𝑛𝑘 , and 𝜆 𝑛 := 1/(𝑛𝑘 − 𝑛𝑘−1 ) for 𝑛𝑘−1 < 𝑛 ≤ 𝑛𝑘 . Then we choose the next natural number 𝑛𝑘+1 such that 𝑛𝑘+1 ≥ 𝑛𝑘 + 𝑘, 𝑛𝑘+1 − 𝑛𝑘 > 1/𝜆 𝑛𝑘 and 𝛼𝑛 < 1/2𝑘+1 for 𝑛 ≥ 𝑛𝑘+1 , and put 𝜆 𝑛 :=

1 𝑛𝑘+1 − 𝑛𝑘

(𝑛𝑘 < 𝑛 ≤ 𝑛𝑘+1 ) .

By construction, for 𝑛𝑘 < 𝑛 ≤ 𝑛𝑘+1 , we then have 𝜆𝑛 =

1 < 𝜆 𝑛𝑘 , 𝑛𝑘+1 − 𝑛𝑘

𝜆𝑛 ≤

1 . 𝑘

2.2 The Waterman variation

| 133

Thus, (𝜆 𝑛 )𝑛 is decreasing and tends to zero. Moreover, we have 𝑛𝑘+1

∑ 𝜆𝑛 = 1

𝑛=𝑛𝑘 +1

which shows that (2.24) is true. On the other hand, 𝑛𝑘+1

𝑛

𝑛𝑘+1

1 𝑘+1 1 1 𝜆𝑛 = 𝑘 ∑ 𝜆𝑛 = 𝑘 , 𝑘 2 2 2 𝑛=𝑛𝑘 +1 𝑛=𝑛𝑘 +1

∑ 𝛼𝑛 𝜆 𝑛 ≤ ∑

𝑛=𝑛𝑘 +1

which proves (2.34). For example, if we take 𝛼𝑛 = 1/𝑛 in Lemma 2.20, then the Waterman sequence Λ = (𝜆 𝑛)𝑛 constructed in the proof has the form Λ = ( 12 , 12 , 14 , 14 , 14 , 14 , 18 , 18 , 18 , 18 , 18 , 18 , 18 , 18 , and (2.34) reads

∞

∞

𝑛=1

𝑘=1

∑ 𝛼𝑛 𝜆 𝑛 ≤ ∑ 2𝑘

1 , . . .) 16

,

∞ 1 1 𝑘 ( ) < ∞. = ∑ 22𝑘 𝑘=1 2

The last auxiliary result we need is based on a combinatorial lemma. Lemma 2.21. Let (𝜆 𝑛)𝑛 be a decreasing sequence of positive real numbers. If (𝛿𝑛 )𝑛 is a sequence of positive real numbers tending to zero, and (𝛿𝑛̂ )𝑛 denotes the sequence (𝛿𝑛 )𝑛 arranged in decreasing order, then 𝑛

𝑛

𝑘=1

𝑘=1

∑ 𝜆 𝑘 𝛿𝑘 ≤ ∑ 𝜆 𝑘 𝛿𝑘̂

(𝑛 = 1, 2, 3, . . .) .

(2.35)

Proof. Fix 𝑛 ∈ ℕ and consider the set {𝛿1 , 𝛿2 , . . . , 𝛿𝑛 }. Let 𝛿1󸀠 ≥ 𝛿2󸀠 ≥ . . . ≥ 𝛿𝑛󸀠 denote the elements of this set arranged in decreasing order. Then from a well-known combina­ torial lemma, we obtain 𝑛

𝑛

𝑘=1

𝑘=1

∑ 𝜆 𝑘 𝛿𝑘 ≤ ∑ 𝜆 𝑘 𝛿𝑘󸀠 .

Clearly,

𝛿𝑘󸀠

≤ 𝛿𝑘̂ for 𝑘 = 1, 2, . . . , 𝑛, and so (2.35) follows.

The next proposition shows that the Waterman space Λ𝐵𝑉 is “stable” under a mono­ tone substitution of variables; compare this with Proposition 1.12. Proposition 2.22. Given a function 𝑔 : [𝑐, 𝑑] → ℝ, let 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] be continuous and strictly increasing with 𝜏(𝑎) = 𝑐 and 𝜏(𝑏) = 𝑑. Then 𝑔 ∘ 𝜏 ∈ Λ𝐵𝑉([𝑎, 𝑏]) if and only if 𝑔 ∈ Λ𝐵𝑉([𝑐, 𝑑]). Proof. Suppose that 𝑔 ∈ Λ𝐵𝑉([𝑐, 𝑑]), and let 𝑆∞ = {[𝑐𝑛 , 𝑑𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑐, 𝑑]). By our assumption on 𝜏, we then have 𝜏(𝑆∞ ) = {[𝜏(𝑐𝑛 ), 𝜏(𝑑𝑛 )] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]). Since

134 | 2 Nonclassical BV-spaces 𝑔 ∈ Λ𝐵𝑉([𝑐, 𝑑]), we have ∞

VarΛ (𝑔 ∘ 𝜏, 𝑆∞ ; [𝑎, 𝑏]) = ∑ 𝜆 𝑛|(𝑔 ∘ 𝜏)(𝑑𝑛 ) − (𝑔 ∘ 𝜏)(𝑐𝑛 )| 𝑛=1 ∞

= ∑ 𝜆 𝑛|𝑔[𝜏(𝑑𝑛 )] − 𝑔[𝜏(𝑐𝑛 )]| = VarΛ (𝑔, 𝜏(𝑆∞ ); [𝑐, 𝑑]) < ∞ , 𝑛=1

which shows that 𝑔 ∘ 𝜏 ∈ Λ𝐵𝑉([𝑎, 𝑏]) after passing to the supremum over 𝑆∞ ∈ 𝛴∞ ([𝑎, 𝑏]). Applying the same reasoning to the function 𝜏−1 : [𝑎, 𝑏] → [𝑐, 𝑑] proves the reverse implication. The next result states that, roughly speaking, every continuous function 𝑓 : [𝑎, 𝑏] → ℝ is contained in some appropriate Waterman space Λ𝐵𝑉([𝑎, 𝑏]); as Example 1.8 shows, this is not true⁷ for the classical space 𝐵𝑉([𝑎, 𝑏]). Proposition 2.23. Every function 𝑓 ∈ 𝐶([𝑎, 𝑏]) is contained in Λ𝐵𝑉([𝑎, 𝑏]) for some Wa­ terman sequence Λ. Proof. Note that the assertion follows from the second equality in (2.33); however, we give a direct proof which gives some insight into the method. Let 𝑓 : [𝑎, 𝑏] → ℝ be continuous; we construct a Waterman sequence Λ = (𝜆 𝑛)𝑛 such that 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]). For any 𝛿 > 0, we denote by 𝜔∞ (𝑓; 𝛿) the modulus of continuity of 𝑓 defined in (0.97). We know that 𝜔∞ (𝑓; ⋅) is increasing and 𝜔∞ (𝑓; 𝛿) → 0 as 𝛿 → 0 since 𝑓 is uniformly continuous on [𝑎, 𝑏]. Let 𝐼𝑛 = [𝑎𝑛 , 𝑏𝑛 ] be a sequence of nonoverlapping subintervals of [𝑎, 𝑏]; as in the proof of Proposition 2.18, we use the shortcut |𝑓(𝐼𝑛 )| := |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|. For 𝑚 ∈ ℕ, put 𝐸𝑚 := {𝐼𝑘 : 𝜔∞ (𝑓;

𝑏−𝑎 𝑏−𝑎 ) < |𝑓(𝐼𝑘 )| ≤ 𝜔∞ (𝑓; )} . 𝑚+1 𝑚

Observe that in case |𝑏𝑘 − 𝑎𝑘 | ≤ (𝑏 − 𝑎)/𝑚, we have |𝑓(𝐼𝑘 )| = |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| ≤ 𝜔∞ (𝑓; |𝑏𝑘 − 𝑎𝑘 |) ≤ 𝜔∞ (𝑓;

𝑏−𝑎 ), 𝑚+1

and so 𝐼𝑘 ∈ 𝐸𝑚 only if |𝑏𝑘 −𝑎𝑘 | > (𝑏−𝑎)/(𝑚+1). Since the intervals 𝐼𝑘 are nonoverlapping and contained in [𝑎, 𝑏], we deduce that 𝐸𝑚 contains 𝑚 intervals at most. Moreover, if 𝐼𝑝 ∈ 𝐸𝑟 and 𝐼𝑞 ∈ 𝐸𝑟+𝑠 , then |𝑓(𝐼𝑞 )| ≤ 𝜔∞ (𝑓;

𝑏−𝑎 𝑏−𝑎 ) ≤ 𝜔∞ (𝑓; ) < |𝑓(𝐼𝑝 )| . 𝑟+𝑠 𝑟+1

7 Note that it is not true either that the whole space 𝐶([𝑎, 𝑏]) is contained in every Waterman space Λ𝐵𝑉([𝑎, 𝑏]), as the first equality in (2.33) shows. That is, for every individual continuous function 𝑓, we have to find an appropriate Waterman sequence Λ such that 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]).

2.2 The Waterman variation

| 135

Thus, we can choose, step by step, a sequence (𝐽𝑛 )𝑛 of intervals 𝐽𝑘 ∈ 𝐸𝑘 satisfying |𝑓(𝐽1 )| ≥ |𝑓(𝐽2 )| ≥ . . . ≥ |𝑓(𝐽𝑛 )| ≥ . . . → 0

(𝑛 → ∞) .

Now, we are going to show that |𝑓(𝐽𝑛 )| ≤ 𝜔∞ (𝑓;

𝑏−𝑎 ) 𝑛

(𝑛 = 1, 2, 3, . . .) .

(2.36)

To this end, assume that 𝑚 ∈ ℕ satisfies 𝑏−𝑎 ); 𝑚

|𝑓(𝐽𝑚 )| > 𝜔∞ (𝑓; then,

|𝑓(𝐽1 )| ≥ |𝑓(𝐽2 )| ≥ . . . ≥ |𝑓(𝐽𝑚 )| > 𝜔∞ (𝑓;

𝑏−𝑎 ). 𝑚

This implies that |𝐽𝑘 | > (𝑏 − 𝑎)/𝑚 for 𝑘 = 1, 2, . . . , 𝑚, which is impossible since all the intervals 𝐽1 , 𝐽2 , . . . , 𝐽𝑚 are nonoverlapping and contained in [𝑎, 𝑏]. Therefore, we have proved (2.36). Furthermore, since 𝜔∞ (𝑓; (𝑏 − 𝑎)/𝑘) → 0 as 𝑘 → ∞, we may apply Lemma 2.20 to the particular choice 𝛼𝑛 := 𝜔∞ (𝑓;

𝑏−𝑎 ) 𝑛

(𝑛 = 1, 2, 3, . . .)

and find a decreasing sequence Λ = (𝜆 𝑛 )𝑛 of positive real numbers tending to zero and satisfying ∞ ∞ 𝑏−𝑎 ) < ∞. ∑ 𝜆 𝑛 = ∞, ∑ 𝜆 𝑛 𝜔∞ (𝑓; 𝑛 𝑛=1 𝑛=1 Subsequently, we apply Lemma 2.21 to the choice 𝛿𝑛 := |𝑓(𝐼𝑛 )| and obtain ∞

∞

∞

∑ 𝜆 𝑛|𝑓(𝐼𝑛 )| ≤ ∑ 𝜆 𝑛|𝑓(𝐽𝑛 )| ≤ ∑ 𝜆 𝑛𝜔∞ (𝑓;

𝑛=1

𝑛=1

𝑛=1

𝑏−𝑎 ) < ∞. 𝑛

(2.37)

However, this means nothing more than 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]) for Λ = (𝜆 𝑛)𝑛 , and the conclusion follows. We may summarize the contents of Proposition 2.23 as inclusion 𝐶([𝑎, 𝑏]) ⊆ ⋃ Λ𝐵𝑉([𝑎, 𝑏]) ,

(2.38)

Λ

where the union is taken over all Waterman sequences Λ. This complements in some sense the second inclusion in (2.28): the union of all Waterman spaces is “intermedi­ ate” between continuous and bounded functions. We now prove the equalities (2.33) which are more precise than the inclusions (2.28). Proposition 2.24. The equalities (2.33) hold, where the intersection and union are taken over all Waterman sequences Λ.

136 | 2 Nonclassical BV-spaces Proof. Observe first that, by (2.28), we only have to prove the inclusion 𝐵𝑉([𝑎, 𝑏]) ⊇ ⋂ Λ𝐵𝑉([𝑎, 𝑏]) Λ

to get the first equality in (2.33). Fix an arbitrary Waterman sequence Λ = (𝜆 𝑛 )𝑛 and let 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]). We then know that 𝑓 is bounded, and so both numbers 𝑚(𝑓) and 𝑀(𝑓) in (0.61) and (0.62) are finite. Putting⁸ 𝐹(𝑥) :=

𝑓(𝑥) − 𝑚(𝑓) 𝑀(𝑓) − 𝑚(𝑓)

(𝑎 ≤ 𝑥 ≤ 𝑏),

we have 0 ≤ 𝐹(𝑥) ≤ 1 for 𝑎 ≤ 𝑥 ≤ 𝑏, and 𝐹 clearly belongs, as 𝑓 itself, to the space Λ𝐵𝑉([𝑎, 𝑏]). This argument shows that we may assume, without loss of generality, that 0 ≤ 𝑓(𝑥) ≤ 1 (𝑎 ≤ 𝑥 ≤ 𝑏) .

(2.39)

Suppose that 𝑓 ∈ ̸ 𝐵𝑉([𝑎, 𝑏]). By Proposition 1.18, we know that then there exists a point 𝑥 ∈ [𝑎, 𝑏] such that 𝑓 is not of bounded (Jordan) variation on any neighborhood of 𝑥. Choose a finite partition 𝑃1 of the interval [𝑎, 𝑏] such that ∑ |𝑓(𝐼)| ≥ 3 . 𝐼∈𝑃1

The point 𝑥 is either an interior point of a single interval in 𝑃1 , or is an endpoint of two intervals in 𝑃1 . If we remove that single interval, or possibly two intervals from 𝑃1 and denote by 𝑄1 the remaining collection of intervals. Then in view of (2.39), we have ∑ |𝑓(𝐼)| ≥ 1 . 𝐼∈𝑄1

If 𝑄1 contains 𝑞1 intervals, we write 𝑄1 = {𝐼𝑘1 : 𝑘 = 1, 2, . . . , 𝑞1 }. Next, we define 𝜆 1 = 𝜆 2 = . . . = 𝜆 𝑞1 := 1 . Then

𝑞1

∑ 𝜆 𝑘 |𝑓(𝐼𝑘1 )| ≥ 1 .

𝑘=1

This completes the first step in our definition. Suppose that we have constructed 𝑃𝑛 and 𝑄𝑛 in this way; then, we proceed induc­ tively as follows. First of all, note that one or possibly two intervals which we have removed from 𝑃𝑛 to form 𝑄𝑛 create a neighborhood 𝑈𝑛 of 𝑥. Since 𝐹 is not of bounded variation on 𝑈𝑛 , we conclude that there exists a finite partition 𝑃𝑛+1 of 𝑈𝑛 such that ∑ |𝑓(𝐼)| ≥ 3 . 𝐼∈𝑃𝑛+1

8 Here we assume, of course, that 𝑓 is not constant; for constant functions there is nothing to prove.

2.2 The Waterman variation

| 137

Notice that the point 𝑥 is again either an interior point of a single interval con­ tained in 𝑃𝑛+1 or an endpoint of at most two intervals in 𝑃𝑛+1 . If we delete this single, or possibly two, intervals from 𝑃𝑛+1 and denote by 𝑄𝑛+1 the remaining collection of intervals, then, again from (2.39), it follows that ∑ |𝑓(𝐼)| ≥ 1 . 𝐼∈𝑄𝑛+1

If 𝑄𝑛+1 contains 𝑞𝑛+1 intervals, we write 𝑄𝑛+1 = {𝐼𝑘𝑛+1 : 𝑘 = 1, 2, . . . , 𝑞𝑛+1 } and define 𝜆 𝑟𝑛 +1 = 𝜆 𝑟𝑛 +2 = . . . = 𝜆 𝑟𝑛 +𝑞𝑛 :=

1 , 𝑛+1

where 𝑟𝑛 := 𝑞1 + 𝑞2 + . . . + 𝑞𝑛 and 𝑞0 = 0. Then we obtain 𝑞𝑛+1

∑ 𝜆 𝑟𝑛 +𝑘 |𝑓(𝐼𝑘𝑛+1 )| ≥

𝑘=1

1 . 𝑛+1

Next, observe that all intervals of 𝑄𝑛+1 contained in 𝑈𝑛 are pairwise nonoverlap­ ping. Thus, we conclude that 𝑛+1 𝑞𝑖

𝑛+1

𝑖=1 𝑘=1

𝑖=1

∑ ∑ 𝜆 𝑟𝑖−1 +𝑘 |𝑓(𝐼𝑘𝑖 )| ≥ ∑

1 . 𝑖

In this way, we have constructed real numbers {𝜆 𝑘 : 𝑘 = 1, 2, 3, . . .} and nonover­ lapping subintervals {𝐼𝑘𝑛 : 𝑘 = 1, 2, . . . , 𝑞𝑛 ; 𝑛 = 1, 2, . . .} of [𝑎, 𝑏] such that (𝜆 𝑘 )𝑘 is de­ creasing and tends to zero, and ∞ 𝑞𝑖

∞

∑ 𝜆 𝑘 = ∞,

𝑘=1

∑ ∑ 𝜆 𝑟𝑖−1 +𝑘 |𝑓(𝐼𝑘1 )| = ∞ . 𝑖=1 𝑘=1

However, this means precisely that Λ = (𝜆 𝑘 )𝑘 is a Waterman sequence and 𝑓 does not belong to the corresponding space Λ𝐵𝑉([𝑎, 𝑏]), contradicting our assump­ tion. Thus, we have proved the first equality in (2.33). Now, we prove the second equality. More precisely, we show that every function 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]) has left limits at every point in (𝑎, 𝑏); the proof for right limits is anal­ ogous. Given 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]), assume that there exists a point 𝑥 ∈ (𝑎, 𝑏) at which 𝑓 does not have a left limit. This means that 𝑙 := lim inf 𝑓(𝑡) < lim sup 𝑓(𝑡) =: 𝐿 . 𝑡→𝑥−

𝑥→𝑥−

Let 𝛿 := (𝐿−𝑙)/3, and choose sequences (𝑝𝑛)𝑛 and (𝑃𝑛 )𝑛 such that 𝑃1 < 𝑃2 < . . . → 𝑥, 𝑝1 < 𝑝2 < . . . → 𝑥, and 𝑓(𝑝𝑛 ) → 𝑙 ,

𝑓(𝑃𝑛 ) → 𝐿 ,

𝑓(𝑝𝑛 ) ≤ 𝑙 + 𝛿 ,

𝑓(𝑃𝑛 ) ≥ 𝐿 − 𝛿 .

138 | 2 Nonclassical BV-spaces Afterwards, we choose a subsequence (𝑄𝑛 )𝑛 of (𝑃𝑛 )𝑛 and a subsequence (𝑞𝑛 )𝑛 of (𝑝𝑛 )𝑛 such that 𝑞1 < 𝑄1 < 𝑞2 < 𝑄2 < ⋅ ⋅ ⋅ . The intervals [𝑞𝑚 , 𝑄𝑚 ] and [𝑞𝑛 , 𝑄𝑛 ] are then disjoint for 𝑚 ≠ 𝑛 and satisfy |𝑓(𝑄𝑛 ) − 𝑓(𝑞𝑛 )| ≥ 𝑓(𝑄𝑛 ) − 𝑓(𝑞𝑛 ) ≥ (𝐿 − 𝛿) − (𝑙 + 𝛿) ≥ 3𝛿 − 2𝛿 = 𝛿 , which implies that

∞

∞

∑ 𝜆 𝑛 |𝑓(𝑄𝑛 ) − 𝑓(𝑞𝑛 )| ≥ 𝛿 ∑ 𝜆 𝑛 = ∞ .

𝑛=1

𝑛=1

This shows that 𝑓 ∈ ̸ Λ𝐵𝑉([𝑎, 𝑏]), contradicting our assumption. It remains to show that every regular function belongs to Λ𝐵𝑉([𝑎, 𝑏]) for some suit­ able Waterman sequence Λ = (𝜆 𝑛 )𝑛 . To this end, we use the Sierpiński decomposition. By Theorem 0.36, there exist a strictly increasing function 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] and a continuous function 𝑔 : [𝑐, 𝑑] → ℝ such that 𝑓 = 𝑔 ∘ 𝜏. The affine function ℓ given by (0.80) is a strictly increasing homeomorphism between the intervals [𝑐, 𝑑] and [𝑎, 𝑏]. Therefore, ℓ ∘ 𝜏 : [𝑎, 𝑏] → [𝑎, 𝑏] is also strictly increasing, and 𝑔 ∘ ℓ−1 : [𝑎, 𝑏] → ℝ is continuous. From Proposition 2.23, we deduce that 𝑔 ∘ ℓ−1 ∈ Λ𝐵𝑉([𝑎, 𝑏]) for some Wa­ terman sequence Λ. On the other hand, Proposition 2.22 implies that 𝑓 = (𝑔∘ℓ−1 )∘(ℓ∘𝜏) belongs to the space Λ𝐵𝑉([𝑎, 𝑏]). This completes the proof. We show now that in the definition (2.26) of the total Waterman variation, we may re­ strict ourselves to rather special collections 𝑆∞ ∈ 𝛴∞ ([𝑎, 𝑏]). To this end, let us denote 𝑑 by 𝛴∞ ([𝑎, 𝑏]) the family of all infinite collections 𝑆𝑑∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} of nonoverlap­ ping subintervals of [𝑎, 𝑏] with the additional property that the sequence (𝛿𝑛 )𝑛 defined by 𝛿𝑛 := |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| (𝑛 = 1, 2, 3, . . .) (2.40) is decreasing and converges to 0. Before proving the announced result, we need an­ other auxiliary lemma on regular functions, see Section 0.3. Lemma 2.25. Let 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]) be a (finite or infinite) collection of intervals, and let 𝑓 ∈ 𝑅([𝑎, 𝑏]). Then one may rearrange the intervals [𝑎𝑛 , 𝑏𝑛 ] ∈ 𝑆∞ in such a way that |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| ≥ |𝑓(𝑏𝑛+1 ) − 𝑓(𝑎𝑛+1 )| for all 𝑛 ∈ ℕ. Proof. Since 𝑓 is regular on [𝑎, 𝑏], by Theorem 0.36, we may find a strictly increasing function 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] (where 𝑐 := 𝜏(𝑎) and 𝑑 := 𝜏(𝑏)) and a continuous func­ tion 𝑔 : [𝑐, 𝑑] → ℝ such that 𝑓 = 𝑔 ∘ 𝜏. Consider the collection of intervals 𝜏(𝑆∞ ) = {[𝜏(𝑎𝑛 ), 𝜏(𝑏𝑛 )] : 𝑛 ∈ ℕ} which belongs to 𝛴∞ ([𝑐, 𝑑]). The sequence (𝜏(𝑏𝑛 ) − 𝜏(𝑎𝑛 ))𝑛 con­ verges to 0. Since every nonnegative sequence converging to 0 can be arranged in de­ creasing order, we find a sequence (𝑛𝑘 )𝑘 of indices such that (𝜏(𝑏𝑛𝑘 ) − 𝜏(𝑎𝑛𝑘 ))𝑘 is de­ creasing (and obviously also converges to 0). Now, consider the sequence (𝜂𝑘 )𝑘 defined by 𝜂𝑘 := |𝑓(𝑏𝑛𝑘 ) − 𝑓(𝑎𝑛𝑘 )| = |𝑔(𝜏(𝑏𝑛𝑘 )) − 𝑔(𝜏(𝑎𝑛𝑘 ))| .

2.2 The Waterman variation

| 139

From the continuity of 𝑔, it follows that 𝜂𝑘 → 0 as 𝑘 → ∞. Applying now a further rearrangement, we find a sequence (𝑘𝑚 )𝑚 of indices such that (𝜂𝑘𝑚 )𝑚 is decreasing. The corresponding sequence of intervals then meets the requirements of our assertion. 𝑑 We now state the announced proposition in terms of the family 𝛴∞ ([𝑎, 𝑏]) introduced before Lemma 2.25.

Proposition 2.26. Let 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]). Then 𝑑 VarΛ (𝑓; [𝑎, 𝑏]) = sup {VarΛ (𝑓, 𝑆𝑑∞ ; [𝑎, 𝑏]) : 𝑆𝑑∞ ∈ 𝛴∞ ([𝑎, 𝑏])} .

(2.41)

Proof. Denote by Var𝑑Λ (𝑓; [𝑎, 𝑏]) the expression on the right-hand side of (2.41). The 𝑑 inequality Var𝑑Λ (𝑓; [𝑎, 𝑏]) ≤ VarΛ (𝑓; [𝑎, 𝑏]) is clear since 𝛴∞ ([𝑎, 𝑏]) ⊆ 𝛴∞ ([𝑎, 𝑏]), and so we only have to prove the reverse inequality. Fix an arbitrary collection 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]), and consider the sequence (𝛿𝑛 )𝑛 defined by (2.40). Applying Lemma 2.25, we can rearrange the sequence (𝛿𝑛 )𝑛 in decreasing order. In this way, we obtain a sequence (𝛿𝑛̂ )𝑛 which, in view of Lemma 2.21, satisfies 𝑛

𝑛

𝑘=1

𝑘=1

∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| ≤ ∑ 𝜆 𝑘 𝛿𝑘̂ .

However, this implies that VarΛ (𝑓; [𝑎, 𝑏]) ≤ Var𝑑Λ (𝑓; [𝑎, 𝑏]) and finishes the proof. Now, we are going to compare the spaces Λ𝐵𝑉 and 𝑀𝐵𝑉 for two different Waterman sequences Λ = (𝜆 𝑛)𝑛 and 𝑀 = (𝜇𝑛 )𝑛 . This requires a technical definition. Definition 2.27. Given two increasing positive sequences (𝐿 𝑛 )𝑛 and (𝑀𝑛 )𝑛 , we write (𝐿 𝑛 )𝑛 ⪯ (𝑀𝑛 )𝑛 if there exists a constant 𝑐 > 0 such that 𝐿 𝑛 ≤ 𝑐𝑀𝑛 for all 𝑛 ∈ ℕ. If both (𝐿 𝑛 )𝑛 ⪯ (𝑀𝑛 )𝑛 and (𝑀𝑛 )𝑛 ⪯ (𝐿 𝑛 )𝑛 , we write (𝐿 𝑛 )𝑛 ∼ (𝑀𝑛 )𝑛 . ◼ Suppose that Λ = (𝜆 𝑛 )𝑛 and 𝑀 = (𝜇𝑛 )𝑛 are two Waterman sequences. To simplify the notation, for 𝑚, 𝑛 ∈ ℕ with 𝑚 ≤ 𝑛, in the sequel we will use the shortcut 𝑛

𝑛

∑ 𝜆 𝑘 =: 𝜆[𝑚, 𝑛],

∑ 𝜇𝑘 =: 𝜇[𝑚, 𝑛] .

𝑘=𝑚

𝑘=𝑚

(2.42)

Note that the crucial property (2.24) of a Waterman sequence (𝜆 𝑛 )𝑛 means that 𝜆[1, 𝑛] → ∞ as 𝑛 → ∞. Proposition 2.28. Let Λ = (𝜆 𝑛 )𝑛 and 𝑀 = (𝜇𝑛 )𝑛 be two Waterman sequences; then, the following is true. (a) 𝑀𝐵𝑉([𝑎, 𝑏]) ⊆ Λ𝐵𝑉([𝑎, 𝑏]) if and only if (𝜆[1, 𝑛])𝑛 ⪯ (𝜇[1, 𝑛])𝑛 . (b) 𝑀𝐵𝑉([𝑎, 𝑏]) = Λ𝐵𝑉([𝑎, 𝑏]) if and only if (𝜆[1, 𝑛])𝑛 ∼ (𝜇[1, 𝑛])𝑛 . Proof. Suppose there exists a constant 𝑐 > 0 such that 𝑛

𝑛

𝜆[1, 𝑛] = ∑ 𝜆 𝑘 ≤ 𝑐 ∑ 𝜇𝑘 = 𝑐𝜇[1, 𝑛] 𝑘=1

𝑘=1

(𝑛 = 1, 2, . . .) .

(2.43)

140 | 2 Nonclassical BV-spaces Fix an arbitrary function 𝑓 ∈ 𝑀𝐵𝑉([𝑎, 𝑏]) and an infinite collection 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]). Putting 𝛿𝑛 := |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|, in view of Proposition 2.26, we can assume without loss of generality that the sequence (𝛿𝑛 )𝑛 is decreasing and tends to 0. Taking into account the identity 𝑛

𝑛−1

∑ 𝜆 𝑘 𝛿𝑘 = ∑ 𝜆[1, 𝑘](𝛿𝑘 − 𝛿𝑘+1 ) + 𝛿𝑛 𝜆[1, 𝑛] , 𝑘=1

𝑘=1

by our assumptions and (2.43), we obtain 𝑛

𝑛−1

𝑛

∑ 𝜆 𝑘 𝛿𝑘 ≤ 𝑐 ( ∑ 𝜇[1, 𝑘](𝛿𝑘 − 𝛿𝑘+1 ) + 𝛿𝑛 𝜇[1, 𝑘]) = 𝑐 ∑ 𝛿𝑘 𝜇𝑘 . 𝑗=1

𝑘=1

𝑘=1

This shows that 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]) and proves the “if” part in (a). To prove the “only if” part, assume now that (𝜆[1, 𝑛])𝑛 ⪯̸ (𝜇[1, 𝑛])𝑛 . Then there exists a strictly increasing unbounded sequence (𝑛𝑘 )𝑘 of integers which satisfies 𝜆[𝑛𝑘 + 1, 𝑛𝑘+1 ] ≥ 2𝑘 𝜇[𝑛𝑘 + 1, 𝑛𝑘+1 ] , where 𝑛0 := 0. Given 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]), let 𝑓 : [𝑎, 𝑏] → ℝ be a function such that⁹ 1 (𝑖 = 𝑛𝑘 + 1, 𝑛𝑘 + 2, . . . , 𝑛𝑘+1 ) . 𝛿𝑖 := |𝑓(𝑏𝑖 ) − 𝑓(𝑎𝑖 )| = 𝑘 2 𝜇[𝑛𝑘 + 1, 𝑛𝑘+1 ] Observe that the sequence (𝛿𝑛 )𝑛 is then decreasing and tends to 0 as 𝑛 → ∞. Moreover, 𝑛𝑘+1

𝑘

∑ 𝛿𝑖 𝜆 𝑖 = ∑ 𝑖=1

and

𝑗=0

𝜆[𝑛𝑗 + 1, 𝑛𝑗+1 ] 2𝑗 𝜇[𝑛𝑗 + 1, 𝑛𝑗+1 ] 𝑛𝑘+1

𝑘

≥ ∑ 2𝑘−𝑗 = 2𝑘+1 − 1 → ∞

(𝑘 → ∞)

𝑗=0

𝑘

1 ≤ 2. 𝑗 2 𝑗=0

∑ 𝛿𝑖 𝜇𝑖 = ∑ 𝑖=1

This shows that 𝑓 ∈ 𝑀𝐵𝑉([𝑎, 𝑏]), but 𝑓 ∉ Λ𝐵𝑉([𝑎, 𝑏]), and so we have proved (a). The statement (b) is of course an immediate consequence of (a). A particularly simple Waterman sequence is Λ 𝑞 := (𝑛−𝑞 )𝑛 for 0 < 𝑞 ≤ 1. This choice of Λ is so important that we introduce a special notation. Definition 2.29. In case 𝜆 𝑛 = 𝑛−𝑞 for 0 < 𝑞 ≤ 1, we denote the corresponding Water­ man space by Λ 𝑞 𝐵𝑉([𝑎, 𝑏]). So, 𝑓 ∈ Λ 𝑞 𝐵𝑉([𝑎, 𝑏]) if and only if ∞

|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| } < ∞, 𝑛𝑞 𝑛=1

VarΛ 𝑞 (𝑓) = VarΛ 𝑞 (𝑓; [𝑎, 𝑏]) = sup { ∑

9 For example, we may take a suitable zigzag function for 𝑓, see Definition 0.49.

2.2 The Waterman variation

| 141

where the supremum is taken over all collections 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]). In the limit case 𝑞 = 1 (i.e. when (2.24) becomes the harmonic series), this space will be denoted by 𝐻𝐵𝑉([𝑎, 𝑏]) and called the space of functions of harmonic bounded vari­ ation. ◼ Historically, the space 𝐻𝐵𝑉 was the first Waterman-type space which occurred quite naturally in the study of the Fourier series. Formally, we could also identify¹⁰ the Wa­ terman space Λ 0 𝐵𝑉 with 𝐵𝑉. Thus, the spaces Λ 𝑞 𝐵𝑉 are, for 0 < 𝑞 < 1, “intermediate” between 𝐵𝑉 and 𝐻𝐵𝑉. Also, from the definition, it readily follows by a simple calcu­ lation that Λ 𝑝 𝐵𝑉([𝑎, 𝑏]) ⊆ Λ 𝑞 𝐵𝑉([𝑎, 𝑏]) ⊆ 𝐻𝐵𝑉([𝑎, 𝑏])

(𝑝 ≤ 𝑞 < 1) .

(2.44)

This may also be considered as a special case of Proposition 2.28 (a). In fact, for Λ = (𝑛−𝑞 )𝑛 and 𝑀 = (𝑛−𝑝 )𝑛 , we have 𝜆[1, 𝑛] = 1 +

1 1 + ... + 𝑞, 2𝑞 𝑛

𝜇[1, 𝑛] = 1 +

1 1 + ... + 𝑝 , 2𝑝 𝑛

so ([𝜆[1, 𝑛])𝑛 ⪯ (𝜇[1, 𝑛])𝑛 if and only if 𝑘𝑞 ≥ 𝑘𝑝 for all 𝑘, and hence 𝑝 ≤ 𝑞, in accordance with (2.44) and Proposition 2.28 (a). To get an idea of these spaces, let us consider the zigzag function 𝑍𝐶,𝐷 constructed in Definition 0.49. Given a Waterman sequence Λ = (𝜆 𝑛)𝑛 , it is easy to see that ∞

VarΛ (𝑍𝐶,𝐷 ; [0, 1]) = ∑ 𝑑𝑛 𝜆 𝑛 ,

(2.45)

𝑛=1

and so 𝑍𝐶,𝐷 belongs to Λ𝐵𝑉([0, 1]) if and only if the series in (2.45) converges. This simple observation makes it possible to separate the classes Λ 𝑞 𝐵𝑉([𝑎, 𝑏]), i.e. to show that inclusion (2.44) is strict for 𝑝 < 𝑞: Example 2.30. Let 𝑍𝜃 be the special zigzag function (0.93) determined by 𝑑𝑛 = 𝑛−𝜃 (𝜃 > 0), and let 𝜆 𝑛 = 𝑛−𝑞 (0 < 𝑞 ≤ 1) as above. Then it follows from (2.45) that 𝑍𝜃 ∈ Λ 𝑞 𝐵𝑉([0, 1]) if and only if 𝜃 + 𝑞 > 1. Thus, if we choose 𝜃 := 1 − 𝑝, the corresponding zigzag function 𝑓 = 𝑍1−𝑝 belongs to Λ 𝑞 𝐵𝑉([0, 1]) for 𝑞 > 𝑝, but not to Λ 𝑝 𝐵𝑉([0, 1]). However, we can do better: for fixed 𝑝 < 1, the same reasoning shows of course that 𝑓 = 𝑍1−𝑝 satisfies 𝑓 ∈ (⋂ Λ 𝑞 𝐵𝑉([0, 1])) \ Λ 𝑝 𝐵𝑉([0, 1]) . 𝑞>𝑝

In particular, the special zigzag function 𝑍1 belongs to 𝐻𝐵𝑉([0, 1]) \ Λ 𝑝 𝐵𝑉([0, 1]) for 𝑝 < 1, and also to 𝐻𝐵𝑉([0, 1]) \ 𝐵𝑉([0, 1]). ♥

10 In the definition of 𝐵𝑉 (Definition 1.1), we only used finite partitions, while in the definition of Λ𝐵𝑉 (Definition 2.15), we used infinite collections of nonoverlapping intervals. However, Proposition 2.18 shows that this is the same, and so we may consider 𝐵𝑉 as a special case of Λ𝐵𝑉 with 𝜆 𝑛 ≡ 1 for finitely many 𝑛.

142 | 2 Nonclassical BV-spaces In the following proposition, we need the notion of Banach indicatrix introduced in Definition 0.38 in the general setting of regular functions, i.e. functions which only have removable discontinuities or discontinuities of first kind (jumps). Proposition 2.31. Let 𝑓 : [𝑎, 𝑏] → ℝ be a regular function, and let 𝐼𝑓 : ℝ → ℕ0 ∪ {∞} be the Banach indicatrix of 𝑓. Suppose that Λ = (𝜆 𝑛 )𝑛 is a Waterman sequence, and (𝜇𝑛 )𝑛 is some positive increasing sequence such that (𝜇𝑛 )𝑛 ∼ (𝜆[1, 𝑛])𝑛 in the sense of Definition 2.27. Finally, assume that the function 𝜇𝑓 : ℝ → ℝ defined by 𝜇𝑓 (𝑦) := 𝜇𝑛 if 𝑛 = 𝐼𝑓 (𝑦) belongs to 𝐿 1 (ℝ), i.e. ∞

∫ 𝜇𝑓 (𝑦) 𝑑𝑦 < ∞ . −∞

Then 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]). Proof. Define 𝑚(𝑓) and 𝑀(𝑓) as in (0.61) and (0.62), respectively, and consider an arbi­ trary sequence (𝐼𝑛 )𝑛 of nonoverlapping subintervals of the interval [𝑎, 𝑏]. Further, de­ fine a sequence (𝑃𝑛 )𝑛 of functions 𝑃𝑛 : [𝑚(𝑓), 𝑀(𝑓)] → ℝ in such a way that 𝑃𝑛 (𝑦) = 𝜆 𝑛 if the equation 𝑓(𝑥) = 𝑦 has at least one solution in the interval 𝐼𝑛 , where we extended the graph of 𝑓 as described in Definition 0.38. On the other hand, we put 𝑃𝑛 (𝑦) := 0 if the equation 𝑓(𝑥) = 𝑦 has no solution in 𝐼𝑛 . Denoting 𝑚𝑛(𝑓) := inf {𝑓(𝑥) : 𝑥 ∈ 𝐼𝑛 },

𝑀𝑛 (𝑓) := sup {𝑓(𝑥) : 𝑥 ∈ 𝐼𝑛 } ,

we see that 𝑃𝑛 = 𝜆 𝑛𝜒(𝑚𝑛 (𝑓),𝑀𝑛 (𝑓)) , that is, {𝜆 𝑛 𝑃𝑛 (𝑦) = { 0 {

for 𝑦 ∈ (𝑚𝑛(𝑓), 𝑀𝑛 (𝑓)) , for 𝑦 ∈ ̸ (𝑚𝑛(𝑓), 𝑀𝑛 (𝑓)) .

Clearly, the function 𝑃𝑛 is measurable. Moreover, 𝑀(𝑓)

∫ 𝑃𝑛 (𝑦) 𝑑𝑦 = 𝜆 𝑛(𝑀𝑛 (𝑓) − 𝑚𝑛(𝑓)) . 𝑚(𝑓)

Now, for arbitrary 𝑛 ∈ ℕ, we have 𝑛

𝑛

∑ 𝜆 𝑘 |𝑓(𝐼𝑘 )| ≤ ∑ 𝜆 𝑘 (𝑀𝑘 (𝑓) − 𝑚𝑘 (𝑓)) 𝑘=1

𝑘=1 ∞

≤

∫ 𝑓(𝐼1 )∪...∪𝑓(𝐼𝑛 )

𝐼𝑓 (𝑦) 𝑑𝑦 ≤ ∫ 𝜇𝑓 (𝑦) 𝑑𝑦 < ∞ . −∞

Since 𝑛 ∈ ℕ was arbitrary, we see that VarΛ (𝑓; [𝑎, 𝑏]) ≤ ‖𝜇𝑓 ‖𝐿 1 , and the conclusion follows.

2.2 The Waterman variation

| 143

Now, we return to the special Waterman space Λ 𝑞 𝐵𝑉 generated by the sequence Λ 𝑞 = (𝑛−𝑞 ) for 𝑞 ∈ (0, 1]. A comparison of the inclusions (1.72) and (2.44) shows that both scales of spaces {𝑊𝐵𝑉𝑝 : 1 ≤ 𝑝 < ∞} and {Λ 𝑝 𝐵𝑉 : 0 < 𝑝 ≤ 1} are increasing with respect to the index 𝑝. Thus, it is natural to ask whether or not these two scales are related. The following Propositions 2.32 and 2.33 give a complete answer to this question and are taken from the survey paper [250]. As usual, for 1 < 𝑝 < ∞, we denote by 𝑝󸀠 := 𝑝/(𝑝 − 1) the conjugate exponent to 𝑝, see (0.13). Proposition 2.32. For 𝑝 > 1 and

1 𝑝󸀠

< 𝑞 ≤ 1, the continuous imbedding (2.46)

𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) 󳨅→ Λ 𝑞 𝐵𝑉([𝑎, 𝑏]) holds true. Moreover, the inclusion 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊂ Λ 𝑞 𝐵𝑉([𝑎, 𝑏]) is strict.

Proof. For the proof, we use our information about the convergence behavior of the series 𝜁(𝛼, 𝛽) defined in (0.17). Let 𝑓 ∈ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) and Var𝑊 𝑝 (𝑓; [𝑎, 𝑏]) =: 𝑀. Fix 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏]), and put 𝜂𝑘 := |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| for 𝑘 = 1, 2, . . . , 𝑛. We know that 𝑝 𝑝 𝜂1 + 𝜂2 + . . . + 𝜂𝑛𝑝 ≤ 𝑀 , and we must find a bound on 𝜂1 + 𝜂2 2−𝑞 + . . . + 𝜂𝑛 𝑛−𝑞 under the assumption 𝑞𝑝󸀠 > 1, i.e. 𝑝𝑞 > 𝑝 − 1. Applying Hölder’s inequality (0.108) to 𝛼𝑘 := 𝜂𝑘 and 𝛽𝑘 := 𝑘−𝑞 (𝑘 = 1, 2, . . . , 𝑛) yields 𝑛

𝑛 𝜂 𝑝 ∑ 𝑘𝑞 ≤ ( ∑ 𝜂𝑘 ) 𝑘 𝑘=1 𝑘=1

1/𝑝

𝑛

(∑

1/𝑝󸀠

1

𝑞𝑝 𝑘=1 𝑘

󸀠

)

󸀠

≤ 𝑀1/𝑝 𝜁(𝑞𝑝󸀠 , 0)1/𝑝 ,

and so 𝑓 ∈ Λ 𝑞 𝐵𝑉([𝑎, 𝑏]) since 𝜁(𝑞𝑝󸀠 , 0) is finite. Moreover, our calculations show that ‖𝑓‖Λ 𝑞 𝐵𝑉 ≤ 𝜁(𝑞𝑝󸀠 , 0)‖𝑓‖𝑊𝐵𝑉𝑝 , and so (2.46) defines, in fact, a continuous imbedding. To show that the inclusion (2.46) is strict, we assume, without loss of generality, that [𝑎, 𝑏] = [0, 1]. Consider the zigzag function 𝑍𝐶,𝐷 from Definition 0.49, determined by the sequences 𝑐𝑛 := Then

∞

1 , 2𝑛

𝑑𝑛 :=

1 𝑛1−𝑞 log2 (𝑛 + 1)

𝑑𝑛 = 𝜁(1, 2) < ∞ , 𝑞 𝑛=1 𝑛 ∑

(𝑛 = 1, 2, 3, . . .) .

∞

∑ 𝑑𝑝𝑛 = 𝜁(𝑝(1 − 𝑞), 2𝑝) = ∞

𝑛=1

since 𝑝(1 − 𝑞) = 𝑝 − 𝑝𝑞 < 𝑝 − (𝑝 − 1) = 1. This shows that 𝑍𝐶,𝐷 ∈ Λ 𝑞 𝐵𝑉([0, 1]), but 𝑍𝐶,𝐷 ∈ ̸ 𝑊𝐵𝑉𝑝 ([0, 1]).

144 | 2 Nonclassical BV-spaces The last example shows that functions 𝑓 ∈ Λ 𝑞 𝐵𝑉([𝑎, 𝑏])\𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) for a fixed value of 𝑞 > 1/𝑝󸀠 exist. However, we can do much better: functions 𝑓 ∈ ( ⋂ Λ 𝑞 𝐵𝑉([0, 1])) \ 𝑊𝐵𝑉𝑝 ([0, 1])

(2.47)

𝑞>1/𝑝󸀠

exist (Exercise 2.6). Proposition 2.32 gives a complete picture for 𝑝 > 1 and 1 < 𝑞𝑝󸀠 ≤ 𝑝󸀠 , but does not cover the case 𝑞𝑝󸀠 ≤ 1. The next proposition shows that in that case, the inclusion goes in the other direction. Proposition 2.33. For 𝑝 > 1 and 𝑞 ≤

1 , 𝑝󸀠

the continuous imbedding (2.48)

Λ 𝑞 𝐵𝑉([𝑎, 𝑏]) 󳨅→ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) holds true. Moreover, the inclusion Λ 𝑞 𝐵𝑉([𝑎, 𝑏]) ⊂ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) is strict.

Proof. Since the scale of spaces Λ 𝑞 𝐵𝑉 is increasing in 𝑞, see (2.44), we may restrict ourselves to the case 𝑞 = 1/𝑝󸀠 . Thus, let 𝑓 ∈ Λ 𝑞 𝐵𝑉([𝑎, 𝑏]) and VarΛ 𝑞 (𝑓; [𝑎, 𝑏]) =: 𝑀. Fix 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏]), and put 𝜂𝑘 := |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| for 𝑘 = 1, 2, . . . , 𝑛. We know that 𝜂1 + 𝜂2 2−𝑞 + . . . + 𝜂𝑛 𝑛−𝑞 ≤ 𝑀 , 𝑝

𝑝

and we must find a bound on 𝜂1 + 𝜂2 + . . . + 𝜂𝑛𝑝 under the assumption 𝑞𝑝󸀠 = 1, i.e. 𝑝𝑞 = 𝑝 − 1. By renumbering the intervals in 𝑆, if necessary, we may assume that 𝜂1 ≥ 𝜂2 ≥ . . . ≥ 𝜂𝑛 , see Proposition 2.26. Under this additional assumption, we get 𝑘

𝑀≥∑

𝜂𝑗

𝑗𝑞 𝑗=1

𝑘

≥∑ 𝑗=1

𝜂𝑘 𝜂 = 𝑘 𝑝𝑘 = 𝑘1−𝑞 𝜂𝑘 = 𝑘1/𝑝 𝜂𝑘 𝑘𝑞 𝑘

(𝑘 = 1, 2, . . . , 𝑛) .

Consequently, 𝑝

𝑝−1

𝑝𝑞

𝜂𝑘 = 𝜂𝑘 𝜂𝑘 ≤ 𝜂𝑘 𝜂𝑘 ≤ and hence

𝑛

𝑝

𝑀𝑝𝑞 𝑀𝑝−1 𝜂𝑘 = 𝜂 , 𝑞 𝑘 𝑘𝑞 𝑘

𝑛

𝜂𝑘 ≤ 𝑀𝑝−1 𝑀 = 𝑀𝑝 . 𝑘𝑞 𝑘=1

∑ 𝜂𝑘 ≤ 𝑀𝑝−1 ∑

𝑘=1

𝑝 This shows that 𝑓 ∈ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) with Var𝑊 𝑝 (𝑓; [𝑎, 𝑏]) ≤ 𝑀 , and so

‖𝑓‖𝑊𝐵𝑉𝑝 ≤ ‖𝑓‖Λ 𝑞 𝐵𝑉 . The proof of the properness of the inclusion (2.48) is left to the reader (Exer­ cise 2.5). We point out that Proposition 2.32 carries over rather easily to the more general setting of the Wiener–Young space 𝑊𝐵𝑉𝜙 defined by the norm (2.11).

2.2 The Waterman variation

| 145

Proposition 2.34. Let 𝜙 : [0, ∞) → [0, ∞) be some Young function, and let 𝜙∗ denote its conjugate Young function (0.23). Let Λ = (𝜆 𝑛)𝑛 be a Waterman sequence satisfying ∞

∑ 𝜙∗ (𝜆 𝑛) < ∞ .

(2.49)

𝑛=1

Then the inclusion 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) ⊆ Λ𝐵𝑉([𝑎, 𝑏]) holds. Proof. Using Young’s inequality (0.24) between 𝜙 and 𝜙∗ , for any 𝑓 ∈ 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) and any collection 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]), we get 𝑁

𝑁

𝑁

𝑛=1

𝑛=1

∑ 𝜆 𝑛 |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| ≤ ∑ 𝜙(|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) + ∑ 𝜙∗ (𝜆 𝑛) ≤

Var𝑊 𝜙 (𝑓; [𝑎, 𝑏])

𝑛=1

∞

∗

+ ∑ 𝜙 (𝜆 𝑛) < ∞ , 𝑛=1

and hence VarΛ (𝑓; [𝑎, 𝑏]) < ∞. 󸀠

Taking 𝜙(𝑡) = 𝑡𝑝 and 𝜆 𝑛 = 𝑛−𝑞 in Proposition 2.34, we get 𝜙∗ (𝑡) = 𝑡𝑝 (up to constants), and so condition (2.49) reads ∞ 1 ∑ 𝑞𝑝󸀠 < ∞ . (2.50) 𝑛=1 𝑛 This holds precisely for 𝑞𝑝󸀠 > 1, and so we recover Proposition 2.32 as a special case of Proposition 2.34. There is another interesting function class which is related to the Waterman space Λ𝐵𝑉 and is based on the notion of the modulus of variation of a function. First, we need another family of intervals similar to 𝛴([𝑎, 𝑏]) and 𝛴∞ ([𝑎, 𝑏]). Definition 2.35. For 𝑛 ∈ ℕ fixed, denote by 𝛴𝑛 ([𝑎, 𝑏]) the family of all collections 𝑆𝑛 := {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]}

(2.51)

of 𝑛 nonoverlapping subintervals [𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ] of [𝑎, 𝑏]. Given a bounded function 𝑓 : [𝑎, 𝑏] → ℝ, the modulus of variation of 𝑓 is the sequence 𝜈(𝑓) = (𝜈(𝑓)𝑛 )𝑛 defined by 𝑛

𝜈(𝑓)𝑛 = 𝜈(𝑓; [𝑎, 𝑏])𝑛 := sup { ∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| : 𝑆𝑛 ∈ 𝛴𝑛 ([𝑎, 𝑏])} ,

(2.52)

𝑘=1

where the supremum in (2.52) is taken over all collections 𝑆𝑛 of the form (2.51). If 𝜈 := (𝜈𝑛 )𝑛 is an arbitrary increasing and concave¹¹ real sequence, the Chanturiya class V𝜈 = V𝜈 ([𝑎, 𝑏]) is defined as a set of all bounded functions 𝑓 : [𝑎, 𝑏] → ℝ satisfying 𝜈(𝑓)𝑛 = 𝑂(𝜈𝑛 ) (𝑛 → ∞) ,

11 Recall that a sequence (𝜈𝑛 )𝑛 is called concave if 𝜈𝑛+1 + 𝜈𝑛−1 ≤ 2𝜈𝑛 for 𝑛 ≥ 2.

(2.53)

146 | 2 Nonclassical BV-spaces with 𝜈(𝑓)𝑛 given by (2.52). In the particular case of the concave sequence 𝜈𝑟 := (𝑛𝑟 )𝑛 with 0 < 𝑟 ≤ 1, we denote the corresponding Chanturiya class by V𝜈𝑟 = V𝜈𝑟 ([𝑎, 𝑏]). ◼ It follows immediately from Definition 2.35 that 𝜈(𝑓)𝑛 ≤ 𝜈(𝑓)𝑛+1

(𝑛 = 1, 2, . . .) .

Another interesting property of (2.52) is contained in Exercise 2.20. The characteristic 𝜈(𝑓) and the function class V𝜈 have been introduced by Chan­ turiya in [79] and applied to the Fourier series in [80–84]. Roughly speaking, the Chan­ turiya class V𝜈 is modeled on the modulus of variation in a similar way as the gener­ alized Hölder space 𝐿𝑖𝑝𝜔,𝑝 (Definition 0.54) on the modulus of continuity of a func­ tion. We point out that interesting interconnections between these moduli exist. For instance, one may show that, for 𝑓 ∈ 𝐶([𝑎, 𝑏]), 𝜔1 (𝑓; 𝛿) = 𝑂 (𝛿𝜈(𝑓)ent(1/𝛿) )

(𝛿 → 0+) ,

(2.54)

where 𝜔1 (𝑓; 𝛿) denotes the integral modulus of continuity (0.98) of 𝑓, and ent(𝜉) is the integer part of 𝜉 ∈ ℝ+ , see Exercise 2.10. Vice versa, one has 𝜈(𝑓)𝑛 = 𝑜 (𝑛𝜔∞ (𝑓; 1/𝑛))

(𝑛 → ∞) ,

(2.55)

where 𝜔∞ (𝑓; 𝛿) is the classical modulus of continuity (0.97), see Exercise 2.11. The following connection of the Chanturiya class V 𝜈 with the Wiener–Young space 𝑊𝐵𝑉𝜙 and the Waterman space Λ𝐵𝑉 is due to Avdispahić [24, 25]. Proposition 2.36. (a) For any Young function 𝜙, the inclusion 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) ⊆ V𝜈 ([𝑎, 𝑏])

(2.56)

1 𝜈𝑛 := 𝑛𝜙−1 ( ) . 𝑛

(2.57)

holds true for

In particular, 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊆ V𝜈 ([𝑎, 𝑏])

(1 ≤ 𝑝 < ∞) ,

1−1/𝑝

. where 𝜈𝑛 = 𝑛 (b) For any Waterman sequence Λ, the inclusion Λ𝐵𝑉([𝑎, 𝑏]) ⊆ V𝜈 ([𝑎, 𝑏]) holds true for 𝜈𝑛 :=

𝑛 , 𝜆[1, 𝑛]

where 𝜆[1, 𝑛] is given by (2.42). In particular, Λ 𝑞 𝐵𝑉([𝑎, 𝑏]) ⊆ V𝜈𝑞 ([𝑎, 𝑏]) where 𝜈𝑞 = (𝑛𝑞 )𝑛 .

(0 < 𝑞 < 1) ,

(2.58)

2.2 The Waterman variation

| 147

Proof. (a) Fix 𝑓 ∈ 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) and 𝑆𝑛 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴𝑛 ([𝑎, 𝑏]). Applying the Jensen inequality (2.18) to 𝛼𝑘 ≡ 1 and 𝑢𝑘 = |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| for 𝑘 = 1, . . . , 𝑛, we get 𝑛 𝜙(|𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|) 1 1 𝑛 ≤ Var𝑊 𝜙 ( ∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|) ≤ ∑ 𝜙 (𝑓) . 𝑛 𝑘=1 𝑛 𝑛 𝑘=1

Applying 𝜙−1 and multiplying by 𝑛 yields 𝑛 1 −1 1 ∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| ≤ 𝑛𝜙−1 ( Var𝑊 𝜙 (𝑓)) ≤ 𝐶𝑛𝜙 ( ) = 𝐶𝜈𝑛 𝑛 𝑛 𝑘=1

for some constant 𝐶 > 0 depending on 𝑓. Choosing, in particular, 𝜙(𝑢) := |𝑢|𝑝 for 1 ≤ 𝑝 < ∞ in (a), the sequence (𝜈𝑛 )𝑛 in (2.57) becomes 𝜈𝑛 = 𝑛

1 = 𝑛1−1/𝑝 ; 𝑛1/𝑝

in particular, 𝜈𝑛 ≡ 1 for 𝑝 = 1. (b) Fix 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]) and 𝑆∞ = {[𝑎𝑘 , 𝑏𝑘 ] : 𝑘 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]). By definition of the Waterman variation (2.25), for 𝑛 ∈ ℕ fixed, we have 𝜆 1 |𝑓(𝑏1 ) − 𝑓(𝑎1 )| + 𝜆 2 |𝑓(𝑏2 ) − 𝑓(𝑎2 )| + . . . + 𝜆 𝑛 |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| ≤ VarΛ (𝑓) , but also 𝜆 2 |𝑓(𝑏1 ) − 𝑓(𝑎1 )| + 𝜆 3 |𝑓(𝑏2 ) − 𝑓(𝑎2 )| + . . . + 𝜆 1 |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| ≤ VarΛ (𝑓) , 𝜆 3 |𝑓(𝑏1 ) − 𝑓(𝑎1 )| + 𝜆 4 |𝑓(𝑏2 ) − 𝑓(𝑎2 )| + . . . + 𝜆 2 |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| ≤ VarΛ (𝑓) , .. . 𝜆 𝑛|𝑓(𝑏1 ) − 𝑓(𝑎1 )| + 𝜆 1 |𝑓(𝑏2 ) − 𝑓(𝑎2 )| + . . . + 𝜆 𝑛−1 |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| ≤ VarΛ (𝑓) . Summing up all these estimates, we obtain 𝑛

𝑛

𝑛

𝜆[1, 𝑛] ∑ |𝑓(𝑏𝑗 ) − 𝑓(𝑎𝑗 )| = (∑ 𝜆 𝑖 ) ( ∑ |𝑓(𝑏𝑗 ) − 𝑓(𝑎𝑗 )|) ≤ 𝑛 VarΛ (𝑓) , 𝑗=1

𝑖=1

and hence 𝜈(𝑓)𝑛 ≤

𝑗=1

𝑛 VarΛ (𝑓) 𝑛 = 𝑂( ), 𝜆[1, 𝑛] 𝜆[1, 𝑛]

and the conclusion follows. To prove the last assertion, we use the fact that in the special case 𝜆 𝑛 = 𝑛−𝑞 , we have 𝑛 1 1 𝜆[1, 𝑛] = ∑ 𝑞 ≥ 𝑛 𝑞 = 𝑛1−𝑞 (0 < 𝑞 < 1) . 𝑘 𝑛 𝑘=1 This implies that

and the assertion follows.

𝑛 = 𝑂 (𝑛𝑞 ) , 𝜆[1, 𝑛]

148 | 2 Nonclassical BV-spaces Table 2.2. Imbeddings into Chanturiya classes. The space

is contained in

for 𝜈 = (𝜈𝑛 )𝑛 with

𝐵𝑉([𝑎, 𝑏])

V𝜈 ([𝑎, 𝑏])

𝜈𝑛 ≡ 1

𝑊𝐵𝑉𝑝 ([𝑎, 𝑏])

V𝜈 ([𝑎, 𝑏])

𝜈𝑛 = 𝑛1−1/𝑝

𝑊𝐵𝑉𝜙 ([𝑎, 𝑏])

V𝜈 ([𝑎, 𝑏])

Λ𝐵𝑉([𝑎, 𝑏])

V𝜈 ([𝑎, 𝑏])

Λ 𝑞 𝐵𝑉([𝑎, 𝑏])

V𝜈 ([𝑎, 𝑏])

1 𝜈𝑛 = 𝑛𝜙−1 ( ) 𝑛 𝑛 𝜈𝑛 = 𝜆1 + . . . + 𝜆𝑛 𝜈𝑛 = 𝑛𝑞

From Proposition 2.36, we obtain a series of imbedding theorems for the Wiener space 𝑊𝐵𝑉𝑝 and the classical space 𝐵𝑉. We summarize these imbeddings in the following Table 2.2. Observe that in the terminology of the last part of Definition 2.35, the first and second rows in Table 2.2 mean that 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊆ V𝜈𝑞 ([𝑎, 𝑏])

(1 ≤ 𝑝 < ∞, 𝑞 = 1 − 1/𝑝) .

Moreover, for this choice of 𝑝 and 𝑞, we deduce from Proposition 2.33 the strict inclusion Λ 𝑞 𝐵𝑉([𝑎, 𝑏]) ⊂ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊆ V𝜈𝑞 ([𝑎, 𝑏]) since 𝑝󸀠 𝑞 = 1. In this sense, the second row in Table 2.2 is sharper than the last row. Yet another generalization of such an imbedding which contains all imbeddings in Proposition 2.36 may be found in Proposition 2.85 in Section 2.8. The following subclass of the Waterman space Λ𝐵𝑉 is important in the theory of summability of the Fourier series, see Section 7.2. Definition 2.37. For 𝑚 ∈ ℕ, the 𝑚-shift of a Waterman sequence Λ = (𝜆 𝑛 )𝑛 is defined by Λ𝑚 := (𝜆 𝑚+𝑛 )𝑛 . A function 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]) is called continuous in Waterman varia­ tion if lim VarΛ𝑚 (𝑓; [𝑎, 𝑏]) = 0 . (2.59) 𝑚→∞

𝑐

We write Λ 𝐵𝑉([𝑎, 𝑏]) for the set of all functions 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]) which are contin­ uous in Waterman variation. ◼ The concept of continuity in Waterman variation was introduced by Waterman [315, 316] under the name continuous Λ-variation, see also [25]. In the following proposition, we give a sufficient condition for continuity in Waterman variation due to Avdispahić [24, 25] in terms of the modulus of variation (2.52) of a bounded function. Proposition 2.38. Let Λ = (𝜆 𝑛)𝑛 be a Waterman sequence and 𝑓 ∈ 𝐵([𝑎, 𝑏]). Suppose that ∞

∑ (𝜆 𝑘 − 𝜆 𝑘+1 )𝜈(𝑓)𝑘 < ∞ ,

(2.60)

𝑘=1

where 𝜈(𝑓) = (𝜈(𝑓)𝑛 )𝑛 denotes the modulus of variation (2.52) of 𝑓. Then 𝑓 ∈ Λ𝑐 𝐵𝑉([𝑎, 𝑏]).

2.2 The Waterman variation

| 149

Proof. Let 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]). By Abel’s partial summation and the fact that (𝜆 𝑛 )𝑛 is decreasing and converges to zero, for fixed 𝑚 ∈ ℕ, we obtain 𝑛

∑ 𝜆 𝑛+𝑚 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| 𝑘=1 𝑛−1

𝑘

𝑛

= ∑ (𝜆 𝑘+𝑚 − 𝜆 𝑘+𝑚+1 ) ∑ |𝑓(𝑏𝑖 ) − 𝑓(𝑎𝑖 )| + 𝜆 𝑛+𝑚 ∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| 𝑖=1

𝑘=1

𝑘=1

𝑛−1

∞

≤ ∑ (𝜆 𝑘+𝑚 − 𝜆 𝑘+𝑚+1 )𝜈(𝑓)𝑘 + 𝜈(𝑓)𝑛 ∑ (𝜆 𝑘 − 𝜆 𝑘+1 ) 𝑘=1

𝑘=𝑛+𝑚

𝑛−1

∞

≤ ∑ (𝜆 𝑘+𝑚 − 𝜆 𝑘+𝑚+1 )𝜈(𝑓)𝑘+𝑚 + ∑ (𝜆 𝑘 − 𝜆 𝑘+1 )𝜈(𝑓)𝑘 𝑘=1 ∞

𝑘=𝑛+𝑚

= ∑ (𝜆 𝑗 − 𝜆 𝑗+1 )𝜈(𝑓)𝑗 . 𝑗=𝑚+1

However, the last expression tends to 0 as 𝑚 → ∞, by (2.60), and thus also 𝑛

∑ 𝜆 𝑛+𝑚 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| → 0 (𝑚 → ∞) .

𝑘=1

Taking the supremum over all collections 𝑆∞ ∈ 𝛴∞ ([𝑎, 𝑏]), we conclude that (2.59) holds. So far, we have considered imbeddings of Waterman spaces into Chanturiya classes. Proposition 2.38 allows us to derive several imbedding theorems in the opposite direc­ tion. We collect some of them in the following Proposition 2.39. (a) Suppose that

∞

𝜈𝑘 1. In Proposition 2.32, we have seen that the same condition on 𝑝 and 𝑞 ensures the inclusion 𝑊𝐵𝑉𝑝 ⊆ Λ 𝑞 𝐵𝑉. However, since Λ𝑐𝑞 𝐵𝑉 is a strict subspace of Λ 𝑞 𝐵𝑉, Proposition 2.39 (b) is better than Proposition 2.32. Our previous discussion shows that for 0 < 𝑞 < 𝑟 < 1, the especially interesting chain of inclusions Λ 𝑞 𝐵𝑉([𝑎, 𝑏]) ⊆ 𝑊𝐵𝑉1/(1−𝑞) ([𝑎, 𝑏]) ⊆ V𝜈𝑞 ([𝑎, 𝑏]) ⊆ Λ𝑐𝑟 𝐵𝑉([𝑎, 𝑏])

(2.65)

2.2 The Waterman variation

| 151

holds, where 𝑊𝐵𝑉𝑝 denotes the Wiener space from Definition 1.31, and 𝜈𝑞 = (𝑛𝑞 )𝑛 as before. These inclusions show that the Wiener space 𝑊𝐵𝑉𝑝 and the Chanturiya class V𝜈 are, for appropriate choices of 𝑝 and 𝜈, in a certain sense, “intermediate” spaces between the Waterman space Λ 𝑞 𝐵𝑉 and Λ𝑐𝑟 𝐵𝑉 which is “slightly larger” than Λ 𝑞 𝐵𝑉 for 𝑟 > 𝑞. In fact, the first inclusion in (2.65) has been proved in Proposition 2.33, the second one follows from Proposition 2.36 (a), and the third one is a consequence of Proposi­ tion 2.39 (a) since ∞ ∞ 𝜈𝑘 1 ∑ 1+𝑟 = ∑ 1+𝑟−𝑞 < ∞ 𝑘 𝑘 𝑘=1 𝑘=1 precisely in case 𝑟 > 𝑞. In Exercise 2.5, we suggest how to show that the first inclusion in (2.65) (or (2.48) is strict. The following two examples also show that the other inclusions in (2.65) are strict. Example 2.40. Let 𝑓 : [0, 1] → ℝ be defined, for 𝑛 = 1, 2, 3, . . ., by 0 { { { { 1 { { 1−𝑞 𝑓(𝑥) := { 𝑛 { {0 { { { {linear

if 𝑥 = 0 , if 𝑥 = if 𝑥 =

1 2𝑛−1 3 2𝑛+1

, ,

otherwise.

We claim that 𝑓 ∈ V𝜈𝑞 ([0, 1]) \ 𝑊𝐵𝑉1/(1−𝑞) ([0, 1]). In fact, choosing the intervals 3 1 [𝑎1 , 𝑏1 ] := [ 34 , 1] , [𝑎2 , 𝑏2 ] := [ 38 , 12 ] , . . . , [𝑎𝑛 , 𝑏𝑛 ] := [ 2𝑛+1 , 2𝑛−1 ],

on which the function 𝑓 is increasing linearly, we have {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴𝑛 ([𝑎, 𝑏]) and¹² 𝑛 1 𝜈(𝑓)𝑛 = ∑ = 𝑂(𝑛𝑞 ) (0 < 𝑞 < 1) , (2.66) (𝑘 + 1)1−𝑞 𝑘=1 by definition of 𝑓, and so 𝑓 ∈ V𝜈𝑞 ([0, 1]). On the other hand, taking the (1 − 𝑞)-th root of the terms after the sum in (2.66), we essentially get the 𝑛-th partial sum of the harmonic series, and so 𝑓 ∈ ̸ 𝑊𝐵𝑉1/(1−𝑞) ([0, 1]). ♥ Example 2.41. To construct a function 𝑓 ∈ Λ𝑐𝑟 𝐵𝑉([0, 1]) \ V𝜈𝑞 ([0, 1]) for 𝑟 > 𝑞, recall that 𝑓 ∈ Λ𝑐𝑟 𝐵𝑉([0, 1]) means that ∞

|𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| } → 0 (𝑚 → ∞) , (𝑘 + 𝑚)𝑟 𝑘=1

sup { ∑

12 The relation (2.66) may be proved by means of the integral comparison test for series.

(2.67)

152 | 2 Nonclassical BV-spaces where the supremum in (2.67) is taken over all infinite collections {[𝑎𝑘 , 𝑏𝑘 ] : 𝑘 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]), while 𝑓 ∈ V𝜈𝑞 ([0, 1]) means that 𝑛

sup { ∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|} = 𝑂(𝑛𝑞 ) ,

(2.68)

𝑘=1

where the supremum in (2.68) is taken over all collections {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴𝑛 ([𝑎, 𝑏]). Thus, if we choose 𝑓 in such a way that the supremum in (2.68) is 𝑂(𝑛𝑝 ) for 𝑞 < 𝑝 < 𝑟, then 𝑓 ∈ Λ𝑐𝑟 𝐵𝑉([0, 1]), but 𝑓 ∈ ̸ V𝜈𝑞 ([0, 1]). ♥ We will summarize several other imbedding theorems in Table 2.7 in Section 2.8.

2.3 The Schramm variation In this section, we discuss the most general concept of variation which contains many of the previously considered concepts as special cases. Unfortunately, the price we have to pay for this generality is that the constructions and proofs become extremely technical and cumbersome. The following is a generalization of Definition 2.15. Definition 2.42. A Schramm sequence is a decreasing sequence 𝛷 = (𝜙𝑛 )𝑛 of Young functions 𝜙𝑛 : [0, ∞) → [0, ∞) (Definition 2.1) such that ∞

∑ 𝜙𝑛 (𝑡) = ∞ (𝑡 > 0) .

(2.69)

𝑛=1

Given a function 𝑓 : [𝑎, 𝑏] → ℝ, a collection 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]), and a Schramm sequence 𝛷 = (𝜙𝑛 )𝑛 , the positive real number ∞

Var𝛷 (𝑓, 𝑆∞ ) = Var𝛷 (𝑓, 𝑆∞ ; [𝑎, 𝑏]) := ∑ 𝜙𝑘 (|𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|)

(2.70)

𝑘=1

is called the Schramm variation of 𝑓 on [𝑎, 𝑏] with respect to 𝑆∞ , while the (possibly infinite) number Var𝛷 (𝑓) = Var𝛷 (𝑓; [𝑎, 𝑏]) := sup {Var𝛷 (𝑓, 𝑆∞ ; [𝑎, 𝑏]) : 𝑆∞ ∈ 𝛴∞ ([𝑎, 𝑏])} ,

(2.71)

where the supremum is taken over all collections 𝑆∞ ∈ 𝛴∞ ([𝑎, 𝑏]), is called the total Schramm variation of 𝑓 on [𝑎, 𝑏]. If Var𝛷 (𝑐𝑓; [𝑎, 𝑏]) < ∞

(2.72)

for some 𝑐 > 0, we say that 𝑓 has bounded Schramm variation on [𝑎, 𝑏] (or bounded 𝛷-variation in Schramm’s sense) and write 𝑓 ∈ 𝛷𝐵𝑉([𝑎, 𝑏]). ◼ As we have done in Definition 1.21 for the space 𝐴𝐶 of absolutely continuous functions, we may also show here that in Definition 2.42, we get the same space 𝛷𝐵𝑉 if in (2.71)

2.3 The Schramm variation

| 153

we restrict ourselves to finite collections 𝑆 ∈ 𝛴([𝑎, 𝑏]). This means that Var𝛷 (𝑓; [𝑎, 𝑏]) = sup {Var𝛷 (𝑓, 𝑆; [𝑎, 𝑏]) : 𝑆 ∈ 𝛴([𝑎, 𝑏])} 𝑛

= sup { ∑ 𝜙𝑘 (|𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|) : {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏])} . 𝑘=1

(2.73) Definition 2.42 is so general that it contains many of the previously discussed spaces which may be obtained by special choices of 𝛷 = (𝜙𝑛 )𝑛 . For further reference, we collect these spaces in the following Proposition 2.43. The spaces 𝐵𝑉, 𝑊𝐵𝑉𝑝 , 𝑊𝐵𝑉𝜙 , and Λ𝐵𝑉 are all of type 𝛷𝐵𝑉 for a suitable choice of 𝛷. Proof. In the same order as in the assertion, for 𝑛 = 1, 2, 3, . . ., take 𝜙𝑛 (𝑡) = 𝑡, 𝜙𝑛 (𝑡) = 𝑡𝑝 (𝑝 ≥ 1), 𝜙𝑛 (𝑡) ≡ 𝜙(𝑡), and 𝜙𝑛 (𝑡) = 𝜆 𝑛𝑡, respectively. As for the space 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]), we may consider the set 𝐵(𝛷) := {𝑓 ∈ 𝐵([𝑎, 𝑏]) : Var𝛷 (𝑓; [𝑎, 𝑏]) ≤ 1}

(2.74)

together with the corresponding Minkowski functional ‖𝑓‖𝛷𝐵𝑉 := |𝑓(𝑎)| + inf {𝜆 > 0 : 𝑓/𝜆 ∈ 𝐵(𝛷)} ,

(2.75)

which is a norm on the space 𝛷𝐵𝑉([𝑎, 𝑏]) = span 𝐵(𝛷). Moreover, the set (2.74) coin­ cides then with the closed unit ball with respect to this norm. We summarize some properties of the space 𝛷𝐵𝑉([𝑎, 𝑏]) with the following Proposition 2.44. The number (2.75) and the set 𝛷𝐵𝑉([𝑎, 𝑏]) have the following proper­ ties. (a) For 𝑓 ≠ 0, we have the estimate Var𝛷 (

𝑓 ) ≤ 1. ‖𝑓‖𝛷𝐵𝑉

(2.76)

(b) From ‖𝑓‖𝛷𝐵𝑉 ≤ 1, it follows that Var𝛷 (𝑓) ≤ ‖𝑓‖𝛷𝐵𝑉 . (c) The set 𝛷𝐵𝑉([𝑎, 𝑏]) is a linear space, and (2.75) defines a norm on 𝛷𝐵𝑉([𝑎, 𝑏]). (d) Convergence in the norm (2.75) implies convergence in the norm (0.39), i.e. uniform convergence on [𝑎, 𝑏]. (e) (𝛷𝐵𝑉([𝑎, 𝑏]), ‖ ⋅ ‖𝛷𝐵𝑉 ) is a Banach space. Proof. By our previous discussion, in all calculations, we may restrict ourselves to fi­ nite collections 𝑆 ∈ 𝛴([𝑎, 𝑏]). The assertions (a) and (b) follow from our general results on Minkowski functionals. To prove (c), we may restrict ourselves without loss of gen­ erality to functions from the subspace 𝛷𝐵𝑉𝑜 ([𝑎, 𝑏]) := {𝑓 ∈ 𝛷𝐵𝑉([𝑎, 𝑏]) : 𝑓(𝑎) = 0}

(2.77)

154 | 2 Nonclassical BV-spaces because otherwise we may pass from the function 𝑓 to the function 𝑓 − 𝑓(𝑎). Clearly, the zero function has norm 0. Thus, let 𝑥 ∈ [𝑎, 𝑏] be such that 𝑓(𝑥) ≠ 0. Then 𝑓 |𝑓(𝑥)| ) → ∞ (𝜆 → 0) . Var𝛷 ( ) ≥ 𝜙1 ( 𝜆 𝜆 Thus, there is a 𝜆 > 0 such that Var𝛷 (𝑓/𝜆) > 1, implying that ‖𝑓‖𝛷𝐵𝑉 ≠ 0. The homogeneity of (2.75) follows from 𝜇𝑓 |𝜇𝑓| ) ≤ 1} = inf {𝜆 > 0 : Var𝛷 ( ) ≤ 1} 𝜆 𝜆 |𝑓| = |𝜇| inf {𝜈 > 0 : Var𝛷 ( ) ≤ 1} = |𝜇|‖𝑓‖𝛷𝐵𝑉 𝜈

‖𝜇𝑓‖𝛷𝐵𝑉 = inf {𝜆 > 0 : Var𝛷 (

for 𝑓 ∈ 𝛷𝐵𝑉𝑜 ([𝑎, 𝑏]), 𝜈 := 𝜆/|𝜆|, and 𝜇 ∈ ℝ\{0}, while the subadditivity of (2.75) follows from the estimate¹³ 𝑛

|(𝑓 + 𝑔)(𝑏𝑘 ) − (𝑓 + 𝑔)(𝑎𝑘 )| ) ‖𝑓 + 𝑔‖𝛷𝐵𝑉

∑ 𝜙𝑘 ( 𝑘=1 𝑛

≤ ∑[ 𝑘=1

‖𝑓‖𝛷𝐵𝑉 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| 𝜙 ( )] ‖𝑓 + 𝑔‖𝛷𝐵𝑉 𝑘 ‖𝑓‖𝛷𝐵𝑉

𝑛

+∑[ 𝑘=1

‖𝑔‖𝛷𝐵𝑉 |𝑔(𝑏𝑘 ) − 𝑔(𝑎𝑘 )| 𝜙 ( )] ≤ 1 ‖𝑓 + 𝑔‖𝛷𝐵𝑉 𝑘 ‖𝑔‖𝛷𝐵𝑉

which holds for 𝑓, 𝑔 ∈ 𝛷𝐵𝑉𝑜 ([𝑎, 𝑏]) and any collection 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛]} ∈ 𝛴([𝑎, 𝑏]). Now, we prove (d). Suppose that 𝑓 ∈ 𝛷𝐵𝑉𝑜 ([𝑎, 𝑏]) satisfies ‖𝑓‖𝛷𝐵𝑉 ≤ 𝜀 for some 𝜀 > 0. Then Var𝛷 (𝑓/𝜀) ≤ ‖𝑓(𝜀‖𝛷𝐵𝑉 ≤ 1 by (b). Consequently, 𝜙1 (

|𝑓(𝑥)| 𝑓 ) ≤ Var𝛷 ( ) ≤ 1 𝜀 𝜀

for all 𝑥 ∈ [𝑎, 𝑏], and so ‖𝑓‖∞ ≤ 𝜀𝜙1−1 (1), where ‖ ⋅ ‖∞ denotes the norm (0.39). It remains to show that the space 𝛷𝐵𝑉𝑜 ([𝑎, 𝑏]) with the norm (2.75) is complete. Let (𝑓𝑚 )𝑚 be a Cauchy sequence in 𝛷𝐵𝑉𝑜 ([𝑎, 𝑏]) with respect to the norm (2.75). By (d), (𝑓𝑚 )𝑚 is then a Cauchy sequence in 𝐵([𝑎, 𝑏]) with respect to the norm (0.39). Since (𝐵([𝑎, 𝑏]), ‖⋅‖∞ ) is complete, there is a function 𝑓 ∈ 𝐵([𝑎, 𝑏]) such that 𝑓𝑚 → 𝑓, as 𝑚 → ∞, uniformly on [𝑎, 𝑏]. Now, given any collection 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏]), choose 𝑛0 ∈ ℕ such that ‖𝑓𝑝 − 𝑓𝑞 ‖𝛷𝐵𝑉 ≤ 𝜀 for 𝑝, 𝑞 ≥ 𝑛0 . Then 𝑛

∑ 𝜙𝑘 (

|(𝑓𝑝 − 𝑓)(𝑏𝑘 ) − (𝑓𝑞 − 𝑓)(𝑎𝑘 )| 𝜀

𝑘=1 𝑛

= lim ∑ 𝜙𝑘 ( 𝑞→∞

𝑘=1

13 Here, we use the convexity of 𝜙𝑘 .

)

|(𝑓𝑝 − 𝑓𝑞 )(𝑏𝑘 ) − (𝑓𝑝 − 𝑓𝑞 )(𝑎𝑘 )| 𝜀

) ≤ 1.

2.3 The Schramm variation

| 155

However, this implies that Var𝛷 ((𝑓𝑚 − 𝑓)/𝜀) ≤ 1 for 𝑚 ≥ 𝑛0 , 𝑓 ∈ 𝛷𝐵𝑉𝑜 ([𝑎, 𝑏]), and ‖𝑓𝑚 − 𝑓‖𝛷𝐵𝑉 → 0 as 𝑚 → ∞. Proposition 2.44 has an interesting consequence. From (d), it follows that the spaces (𝛷𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]), ‖ ⋅ ‖𝛷𝐵𝑉 ) and (Λ𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]), ‖ ⋅ ‖Λ𝐵𝑉 ) are also Banach spaces. The next proposition gives a necessary and sufficient condition for two Young se­ quences 𝛷 and 𝛹 under which one of the corresponding Schramm spaces is contained in the other. Proposition 2.45. Given two Schramm sequences 𝛷 = (𝜙𝑛 )𝑛 and 𝛹 = (𝜓𝑛 )𝑛 , we have the inclusion 𝛷𝐵𝑉([𝑎, 𝑏]) ⊆ 𝛹𝐵𝑉([𝑎, 𝑏]) (2.78) if

𝑛

𝑛

∑ 𝜙𝑘 (𝑡) ≥ 𝑐 ∑ 𝜓𝑘 (𝑡) 𝑘=1

(0 ≤ 𝑡 ≤ 𝑇; 𝑛 = 1, 2, . . .)

(2.79)

𝑘=1

for some 𝑇 > 0 and 𝑐 > 0. The proof of Proposition 2.45 is elementary and left to the reader (Exercise 2.24). Simple examples show that condition (2.79) is not necessary for the inclusion (2.78) to hold (Exercise 2.25). Now, we are going to prove a counterpart of (2.33) for Schramm spaces. To this end, let us recall that 𝑅([𝑎, 𝑏]) denotes the space of all regular functions on the interval [𝑎, 𝑏], see Section 0.3. Proposition 2.46. The equalities ⋂ 𝛷𝐵𝑉([𝑎, 𝑏]) = 𝐵𝑉([𝑎, 𝑏]), 𝛷

⋃ 𝛷𝐵𝑉([𝑎, 𝑏]) = 𝑅([𝑎, 𝑏])

(2.80)

𝛷

hold, where the intersection and union in (2.80) are taken over all Schramm sequences 𝛷. Proof. We start with the union in (2.80). First, we show that 𝛷𝐵𝑉([𝑎, 𝑏]) ⊆ 𝑅([𝑎, 𝑏])

(2.81)

for an arbitrary Schramm sequence 𝛷 = (𝜙𝑛 )𝑛 . So, given 𝑓 ∈ 𝛷𝐵𝑉([𝑎, 𝑏]), we show that 𝑓 has a left limit at each point 𝑥 ∈ (𝑎, 𝑏]. Suppose that this is false, which means that there exists 𝑥0 ∈ (𝑎, 𝑏] such that 𝑓(𝑥0 −) does not exist, and hence 𝑙 := lim inf 𝑓(𝑥) < lim sup 𝑓(𝑥) =: 𝐿 . 𝑥→𝑥0 −

𝑥→𝑥0 −

Take 𝛿 = (𝐿 − 𝑙)/3. Then, in view of the definition of 𝑙 and 𝐿, we can find strictly increasing sequences (𝑃𝑛 )𝑛 and (𝑄𝑛 )𝑛 satisfying 𝑓(𝑃𝑛 ) ≤ 𝑙 + 𝛿, 𝑓(𝑄𝑛 ) ≥ 𝐿 − 𝛿, and lim 𝑓(𝑃𝑛 ) = 𝑙,

𝑛→∞

lim 𝑓(𝑄𝑛 ) = 𝐿 .

𝑛→∞

156 | 2 Nonclassical BV-spaces Choose a subsequence (𝑝𝑛 )𝑛 of (𝑃𝑛 )𝑛 and a subsequence (𝑞𝑛 )𝑛 of (𝑄𝑛 )𝑛 such that 𝑝1 < 𝑞1 < 𝑝2 < 𝑞2 < . . . < 𝑝𝑛 < 𝑞𝑛 < . . . , and consider the collection {[𝑝𝑛 , 𝑞𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]). Then we have |𝑓(𝑞𝑛 ) − 𝑓(𝑝𝑛 )| ≥ 𝑓(𝑞𝑛 ) − 𝑓(𝑝𝑛 ) ≥ 𝐿 − 𝛿 − (𝑙 + 𝛿) = 𝛿 , and hence

∞

∞

∑ 𝜙𝑛 (|𝑓(𝑞𝑛 ) − 𝑓(𝑝𝑛 )|) ≥ ∑ 𝜙𝑛 (𝛿) = ∞

𝑛=1

𝑛=1

by (2.69). However, this contradicts our assumption 𝑓 ∈ 𝛷𝐵𝑉([𝑎, 𝑏]). Thus, we have shown that the left limit 𝑓(𝑥0 −) exists at each point 𝑥0 ∈ (𝑎, 𝑏]. The proof for the right limit is similar and is therefore omitted. This completes the proof of the inclu­ sion (2.81). Observe that (2.81) yields the inclusion ⋃ 𝛷𝐵𝑉([𝑎, 𝑏]) ⊆ 𝑅([𝑎, 𝑏]) ,

(2.82)

𝛷

and so in order to prove the second equality in (2.80), we have to show that every regular functions belongs to some Schramm space. However, in (2.33), we have already shown that every regular function belongs to some Waterman space, and combining this with Proposition 2.43, we obtain 𝑅([𝑎, 𝑏]) ⊆ ⋃ Λ𝐵𝑉([𝑎, 𝑏]) ⊆ ⋃ 𝛷𝐵𝑉([𝑎, 𝑏]) , Λ

𝛷

which proves the second equality in (2.80). Now, we proceed with the proof of the first equality in (2.80). First, observe that for an arbitrary fixed Waterman sequence Λ, the inclusion ⋂ 𝛷𝐵𝑉([𝑎, 𝑏]) ⊆ Λ𝐵𝑉([𝑎, 𝑏])

(2.83)

𝛷

holds. Thus, taking the intersection over all Waterman sequences on the right-hand side of (2.83), we obtain ⋂ 𝛷𝐵𝑉([𝑎, 𝑏]) ⊆ ⋂ Λ𝐵𝑉([𝑎, 𝑏]) . 𝛷

Λ

Combining this with (2.33), we get ⋂ 𝛷𝐵𝑉([𝑎, 𝑏]) ⊆ 𝐵𝑉([𝑎, 𝑏]) ,

(2.84)

𝛷

and it remains to prove the reverse inclusion. First, we make some remarks on Schramm sequences. Given a Schramm sequence 𝛷 = (𝜙𝑛 )𝑛 , we know that for each

2.3 The Schramm variation

| 157

𝑛 ∈ ℕ, the function 𝑡 󳨃→ 𝜙𝑛 (𝑡)/𝑡 is increasing on the half-axis (0, ∞). This implies, in particular, that 𝜙𝑛 (𝑡) 𝜙𝑛 (1) ≤ = 𝜙𝑛 (1) 𝑡 1 for 𝑡 ∈ (0, 1]. Consequently, 𝜙𝑛 (𝑡) ≤ 𝜙𝑛 (1)𝑡

(0 ≤ 𝑡 ≤ 1, 𝑛 = 1, 2, . . .) .

Moreover, 𝜙𝑛 (𝑡) ≤ 𝜙1 (𝑡) ≤ 𝜙1 (1)𝑡

(0 ≤ 𝑡 ≤ 1, 𝑛 = 1, 2, . . .)

since the sequence (𝜙𝑛 )𝑛 is decreasing. Now, fix 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) and 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]). Denote by 𝐴 the set of all indices 𝑛 ∈ ℕ for which |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| ≤ 1, and by 𝐵 := ℕ \ 𝐴, the other indices. Then ∞

∑ 𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) = ∑ 𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) + ∑ 𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|)

𝑛=1

𝑛∈𝐵

𝑛∈𝐴

≤ ∑ 𝜙1 (1)|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| + ∑ 𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) 𝑛∈𝐵

𝑛∈𝐴

= 𝜙1 (1) ∑ |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| + ∑ 𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) 𝑛∈𝐵

𝑛∈𝐴

≤ 𝜙1 (1) Var(𝑓; [𝑎, 𝑏]) + ∑ 𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) . 𝑛∈𝐵

However, the last sum in these estimates contains at most 𝑁 := ent(Var(𝑓; [𝑎, 𝑏])) terms, where ent(𝜉) denotes the integer part of 𝜉. Otherwise, this sum would have at least 𝑁 + 1 terms and we would get Var(𝑓; [𝑎, 𝑏]) > ∑ |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| ≥ 𝑁 + 1 = ent(Var(𝑓; [𝑎, 𝑏])) + 1 , 𝑛∈𝐵

that is, a contradiction. Thus, we obtain ∑ 𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) ≤ ∑ 𝜙𝑛 (2‖𝑓‖∞ )

𝑛∈𝐵

𝑛∈𝐵

≤ ∑ 𝜙1 (2‖𝑓‖∞ ) ≤ 𝜙1 (2‖𝑓‖∞ ) Var(𝑓; [𝑎, 𝑏]) , 𝑛∈𝐵

where ‖ ⋅ ‖∞ denotes the norm (0.39). Combining these estimates yields ∞

∑ 𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) ≤ 𝜙1 (1) Var(𝑓; [𝑎, 𝑏]) + 𝜙1 (2‖𝑓‖∞ ) Var(𝑓; [𝑎, 𝑏])

𝑛=1

≤ [𝜙1 (1) + 𝜙1 (2‖𝑓‖∞ )] Var(𝑓; [𝑎, 𝑏]) , and hence Var𝛷 (𝑓; [𝑎, 𝑏]) ≤ [𝜙1 (1) + 𝜙1 (2‖𝑓‖∞ )] Var(𝑓; [𝑎, 𝑏]) .

158 | 2 Nonclassical BV-spaces However, this means precisely that 𝑓 ∈ 𝛷𝐵𝑉([𝑎, 𝑏]), and so we have shown that 𝐵𝑉([𝑎, 𝑏]) ⊆ 𝛷𝐵𝑉([𝑎, 𝑏]) since 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) was arbitrary. However, the Schramm sequence 𝛷 was also chosen arbitrarily. Thus, we conclude that 𝐵𝑉([𝑎, 𝑏]) ⊆ ⋂ 𝛷𝐵𝑉([𝑎, 𝑏]) 𝛷

which together with (2.84), completes the proof of the first equality in (2.80). In the following proposition which is parallel to Proposition 1.7, we compare the con­ tinuity of a function 𝑓 and that of its variation function. Proposition 2.47. Suppose that 𝑓 ∈ 𝛷𝐵𝑉([𝑎, 𝑏]) is continuous at some point 𝑥0 ∈ [𝑎, 𝑏]. Then the Schramm variation function V𝑓,𝛷 : [𝑎, 𝑏] → ℝ defined by V𝑓,𝛷 (𝑥) := Var𝛷 (𝑓; [𝑎, 𝑥])

(𝑎 ≤ 𝑥 ≤ 𝑏)

is also continuous at 𝑥0 . Proof. We restrict ourselves to showing that the variation function is right continuous at 𝑥0 . Fix 𝑥0 ∈ [𝑎, 𝑏), and take a finite collection {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛1 , 𝑏𝑛1 ]} ∈ 𝛴([𝑥0 , 𝑏]) of nonoverlapping subintervals of [𝑥0 , 𝑏] which are arranged in decreasing order. More­ over, we assume that 𝑛1

∑ 𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) >

𝑛=1

1 Var𝛷 (𝑓; [𝑥0 , 𝑏]) . 2

Since 𝑓 is right continuous at 𝑥0 , by assumption, and each 𝜙𝑛 is continuous, we may assume that 𝐼𝑛 ⊂ (𝑥0 , 𝑏] for 𝑛 = 1, 2, . . . , 𝑛1 . Choose a number 𝑦1 ∈ (𝑥0 , 12 (𝑥0 + 𝑏)) in such a way that 𝑛1

[𝑥0 , 𝑦1 ] ∩ ⋃[𝑎𝑛 , 𝑏𝑛 ] = 0 𝑛=1

and |𝑓(𝑑) − 𝑓(𝑐)| ≤ |𝑓(𝑏𝑛1 ) − 𝑓(𝑎𝑛1 )| for any interval [𝑐, 𝑑] ⊂ (𝑥0 , 𝑦1 ). In this way, we obtain a collection of intervals [𝑎𝑛 , 𝑏𝑛 ] ⊂ (𝑥0 , 𝑦1 ] for 𝑛 = 𝑛1 + 1, 𝑛1 + 2, . . . , 𝑛2 which satisfy |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )| ≥ |𝑓(𝑏𝑛+1 ) − 𝑓(𝑎𝑛+1 )| (𝑛 = 𝑛1 + 1, 𝑛1 + 2, . . . , 𝑛2 ) and

𝑛2

∑ 𝜙𝑛−𝑛1 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) >

𝑛=𝑛1 +1

1 Var𝛷 (𝑓; [𝑥0 , 𝑦1 ]) . 2

Proceeding with this process, for 𝑘 = 1, 2, 3, . . . points 𝑦𝑘 and intervals, we obtain [𝑎𝑛𝑘 +1 , 𝑏𝑛𝑘 +1 ], [𝑎𝑛𝑘 +2 , 𝑏𝑛𝑘 +2 ], . . . , [𝑎𝑛𝑘+1 , 𝑏𝑛𝑘+1 ] ⊆ [𝑦𝑘+1 , 𝑦𝑘 ] (𝑘 = 1, 2, . . .) such that the sequence (𝑦𝑘 )𝑘 is decreasing, 𝑦𝑘 → 𝑥0 as 𝑘 → ∞, the sequence (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|)𝑛 is also decreasing, and 𝑛𝑘+1

∑ 𝜙𝑛−𝑛𝑘 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) >

𝑛=𝑛𝑘 +1

1 Var𝛷 (𝑓; [𝑥0 , 𝑦𝑘 ]) . 2

2.3 The Schramm variation

However,

| 159

∞

∑ 𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) < ∞

𝑛=1

since 𝑓 has bounded Schramm variation, and so for a given 𝜀 > 0, we find an index 𝑁 ∈ ℕ such that ∞ 𝜀 ∑ 𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) < . 2 𝑛=𝑁+1 Keeping in mind that the real sequence (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|)𝑛 is decreasing and con­ verges to zero, we deduce that ∞

∑ 𝜙𝑛 (|𝑓(𝑏𝑛+𝑗 ) − 𝑓(𝑎𝑛+𝑗 )|)
0 and 𝛿 > 0, we introduce the numbers 𝑛 } { 𝑘(𝑀, 𝛿) := min {𝑛 ∈ ℕ : ∑ 𝜙𝑗 (𝛿/2) > 𝑀} ∈ ℕ 𝑗=1 } {

and 𝛽(𝑝, 𝛿) := min {1 −

𝜙𝑗 (𝛿/2) 𝜙𝑗 (𝛿)

: 𝑗 = 1, 2, . . . , 𝑝} ∈ ℝ .

(2.85)

(2.86)

Note that 𝑀1 ≥ 𝑀2 implies 𝑘(𝑀1 , 𝛿) ≥ 𝑘(𝑀2 , 𝛿), and 𝑝1 ≥ 𝑝2 implies 𝛽(𝑝1 , 𝛿) ≤ 𝛽(𝑝2 , 𝛿). Moreover, 𝛽(𝑝, 𝛿) > 0 for any 𝑝 ∈ ℕ and 𝛿 > 0. In what follows, we will utilize the following technical lemma for the Schramm variation function introduced in Proposition 2.47.

160 | 2 Nonclassical BV-spaces Lemma 2.48. Let 𝑓 ∈ 𝛷𝐵𝑉([𝑎, 𝑏]) and 𝛿 > 0. Suppose that 𝑎 ≤ 𝑥 < 𝑦 ≤ 𝑏 such that |𝑓(𝑦) − 𝑓(𝑥)| ≥ 𝛿. Then V𝑓,𝛷 (𝑦) − V𝑓,𝛷 (𝑥) ≥ 𝛽0 𝜙𝑘0 (𝛿) , where 𝑘0 := 𝑘(V𝑓,𝛷 (𝑥), 𝛿) and 𝛽0 := 𝛽(𝑘0 , 𝛿) are defined as in (2.85) and (2.86). We do not prove this lemma since the proof is extremely technical (see Lemma 2.5 in [286]). Instead, we use this result to state a Helly-type theorem for sequences in 𝛷𝐵𝑉([𝑎, 𝑏]). Theorem 2.49. Let (𝑓𝑛 )𝑛 be a sequence in 𝛷𝐵𝑉([𝑎, 𝑏]) such that ‖𝑐𝑓𝑛 ‖∞ ≤ 𝑀,

Var𝛷 (𝑐𝑓𝑛 ; [𝑎, 𝑏]) ≤ 𝑀 (𝑛 = 1, 2, . . .)

for some constants 𝑐 > 0 and 𝑀 > 0. Then (𝑓𝑛 )𝑛 contains a subsequence which converges pointwise on [𝑎, 𝑏] to some 𝑓 ∈ 𝛷𝐵𝑉([𝑎, 𝑏]); moreover, Var𝛷 (𝑐𝑓; [𝑎, 𝑏]) ≤ 𝑀. Proof. For 𝑛 = 1, 2, 3, . . ., put 𝑣𝑛 (𝑥) := V𝑐𝑓𝑛 ,𝛷 (𝑥) = Var𝛷 (𝑐𝑓𝑛 ; [𝑎, 𝑥])

(𝑎 ≤ 𝑥 ≤ 𝑏) .

Being monotonically increasing and bounded on [𝑎, 𝑏], each function 𝑣𝑛 belongs to 𝐵𝑉([𝑎, 𝑏]). Thus, we may apply the classical Helly selection principle (Theorem 1.11) and obtain a subsequence (𝑣𝑛𝑘 )𝑘 of the sequence (𝑣𝑛 )𝑛 and a function 𝑣 such that 𝑣𝑛𝑘 (𝑥) → 𝑣(𝑥) for all 𝑥 ∈ [𝑎, 𝑏] as 𝑘 → ∞. Clearly, 𝑣 is increasing on [𝑎, 𝑏], see Exer­ cise 1.17, and satisfies 0 ≤ 𝑣(𝑥) ≤ 𝑀 for 𝑎 ≤ 𝑥 ≤ 𝑏. Using the usual diagonalization procedure, we may find a subsequence (𝑓𝑗 )𝑗 of the sequence (𝑓𝑛𝑘 )𝑘 which converges at the endpoints 𝑎 and 𝑏 as well as at all rational points in [𝑎, 𝑏]. Denote by (𝑣𝑗 )𝑗 the cor­ responding subsequence of (𝑣𝑛𝑘 )𝑘 . Since 𝑣 is increasing, it is continuous on [𝑎, 𝑏] \ 𝑁, where 𝑁 ⊂ [𝑎, 𝑏] is at most countable. Let 𝑥0 ∈ [𝑎, 𝑏) \ 𝑁 be irrational. Then we find a rational number 𝑦 ∈ (𝑥0 , 𝑏) such that 𝛽1 𝜙𝑘1 (𝜀)

, 3 where 𝑘1 := 𝑘(𝑀, 𝜀) and 𝛽1 := 𝛽(𝑘1 , 𝜀) are defined according to (2.85) and (2.86). We can also find an index 𝐽 such that |𝑣𝑗 (𝑦) − 𝑣(𝑦)| < 𝜂 and |𝑣𝑗 (𝑥0 ) − 𝑣(𝑥0 )| < 𝜂 for 𝑗 < 𝐽. For these 𝑗, we obtain¹⁴ 0 ≤ 𝑣(𝑦) − 𝑣(𝑥0 ) < 𝜂 :=

0 ≤ 𝑣𝑗 (𝑦) − 𝑣𝑗 (𝑥0 ) < 3𝜂 = 𝛽1 𝜙𝑘1 (𝜀) ≤ 𝛽0 𝜙𝑘0 (𝜀) , where 𝑘0 := 𝑘(𝑣𝑗 (𝑥0 ), 𝜀) and 𝛽0 = 𝛽(𝑘0 , 𝜀). In view of Lemma 2.48, we have |𝑓𝑗 (𝑦) − 𝑓𝑗 (𝑥0 )| < 𝜀 for 𝑗 > 𝐽. Since the sequence (𝑓𝑗 (𝑦))𝑗 is convergent, there exists an index 𝐽1 such that |𝑓𝑗 (𝑦) − 𝑓𝑘 (𝑦)| < 𝜀 for 𝑗, 𝑘 > 𝐽1 . Thus, we get |𝑓𝑗 (𝑥0 ) − 𝑓𝑘 (𝑥0 )| < 𝜀 (𝑗, 𝑘 > max {𝐽, 𝐽1 }) ,

14 The last inequality is a consequence of the monotonicity of the quantities (2.85) and (2.86) in 𝑀 and 𝑝.

2.4 The Riesz–Medvedev variation

|

161

which implies that the sequence (𝑓𝑗 (𝑥0 ))𝑗 is convergent. Our discussion shows that the sequence (𝑓𝑗 (𝑥))𝑗 fails to converge only for 𝑥 ∈ 𝑁 ∪ [(𝑎, 𝑏) ∩ ℚ], which is a countable set. Applying again a diagonalization procedure, we may find a subsequence (𝑓𝑚 )𝑚 of (𝑓𝑗 )𝑗 which converges for all 𝑥 ∈ [𝑎, 𝑏]; we denote the corresponding limit by 𝑓(𝑥). Let (𝑣𝑚 )𝑚 be the corresponding subsequence of (𝑣𝑗 )𝑗 , and choose an arbitrary col­ lection 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛0 , 𝑏𝑛0 ]} ∈ 𝛴([𝑎, 𝑏]). Then, for any 𝜀 > 0, we may find 𝑚0 ∈ ℕ such that 𝑣(𝑏) > 𝑣𝑚 (𝑏) − 𝜀 and 󵄨󵄨 󵄨󵄨 𝑛0 𝑛0 󵄨 󵄨󵄨 󵄨󵄨 ∑ 𝜙𝑛 (𝑐|𝑓𝑚 (𝑏𝑛 ) − 𝑓𝑚 (𝑎𝑛 )|) − ∑ 𝜙𝑛 (𝑐|𝑓(𝑏) − 𝑓(𝑎)|)󵄨󵄨󵄨 < 𝜀 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑛=1 𝑛=1 for 𝑚 > 𝑚0 . Then, for these 𝑚, we get 𝑛0

𝑀 ≥ 𝑣(𝑏) > 𝑣𝑚 (𝑏) − 𝜀 ≥ ∑ 𝜙𝑛 (𝑐|𝑓𝑚 (𝑏𝑛 ) − 𝑓𝑚 (𝑎𝑛 )|) − 𝜀 𝑛=1

𝑛0

≥ ∑ 𝜙𝑛 (𝑐|𝑓(𝑏) − 𝑓(𝑎)|) − 2𝜀 . 𝑛=1

However, this implies that Var𝛷 (𝑐𝑓; [𝑎, 𝑏]) < 𝑀 + 2𝜀 , and hence Var𝛷 (𝑐𝑓; [𝑎, 𝑏]) ≤ 𝑀 since 𝜀 > 0 was arbitrary, and completes the proof. Choosing, in particular, 𝜙𝑛 (𝑡) = 𝜆 𝑛𝑡 for 𝑡 ≥ 0, where Λ = (𝜆 𝑛 )𝑛 is some Waterman sequence, we obtain a Helly-type selection principle in Λ𝐵𝑉 from Theorem 2.49.

2.4 The Riesz–Medvedev variation In this section, we study yet another concept of variation which goes back to Riesz [264, 265], contains the Jordan variation as special case and has particularly interest­ ing applications. Definition 2.50. Given a real number 𝑝 ≥ 1, a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), and a function 𝑓 : [𝑎, 𝑏] → ℝ, the nonnegative real number 𝑚

Var𝑅𝑝 (𝑓, 𝑃) = Var𝑅𝑝 (𝑓, 𝑃; [𝑎, 𝑏]) := ∑

𝑗=1

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|𝑝 (𝑡𝑗 − 𝑡𝑗−1 )𝑝−1

(2.87)

is called the Riesz variation of 𝑓 on [𝑎, 𝑏] with respect to 𝑃, while the (possibly infinite) number Var𝑅𝑝 (𝑓) = Var𝑅𝑝 (𝑓; [𝑎, 𝑏]) := sup {Var𝑅𝑝 (𝑓, 𝑃; [𝑎, 𝑏]) : 𝑃 ∈ P([𝑎, 𝑏])} ,

(2.88)

162 | 2 Nonclassical BV-spaces where the supremum is again taken over all partitions of [𝑎, 𝑏], is called the total Riesz variation of 𝑓 on [𝑎, 𝑏]. In case Var𝑅𝑝 (𝑓; [𝑎, 𝑏]) < ∞, we say that 𝑓 has bounded Riesz variation (or bounded 𝑝-variation in Riesz’s sense) on [𝑎, 𝑏] and write¹⁵ 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]). ◼ From Hölder’s inequality (0.108), it follows that (𝑏 − 𝑎)1−1/𝑝 Var𝑅𝑝 (𝑓; [𝑎, 𝑏])1/𝑝 ≥ Var(𝑓; [𝑎, 𝑏]) ,

(2.89)

which shows that the inclusion 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊆ 𝐵𝑉([𝑎, 𝑏]) holds. Moreover, similarly as we have done this in Proposition 1.3 (d) and Proposition 1.10 for 𝐵𝑉, we may prove the following result for 𝑅𝐵𝑉𝑝 . Proposition 2.51. The set 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) equipped with the norm ‖𝑓‖𝑅𝐵𝑉𝑝 := |𝑓(𝑎)| + Var𝑅𝑝 (𝑓; [𝑎, 𝑏])1/𝑝

(𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]))

(2.90)

is a Banach space which is for 𝑝 > 1 continuously imbedded into the space 𝐶([𝑎, 𝑏]) with norm (0.45) as well as into the space 𝐵𝑉([𝑎, 𝑏]) with norm (1.16). Moreover, 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) is an algebra with (2.91) Var𝑅𝑝 (𝑓𝑔) ≤ ‖𝑓‖∞ Var𝑅𝑝 (𝑔) + ‖𝑔‖∞ Var𝑅𝑝 (𝑓) for all 𝑓, 𝑔 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]). Finally, the set 𝑅𝐵𝑉𝑜𝑝 ([𝑎, 𝑏]) of all 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) satisfying 𝑓(𝑎) = 0 is a subalgebra. Proof. It is not hard to prove that 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) is a linear space, and that every func­ tion 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) for 𝑝 > 1 is (uniformly) continuous on [𝑎, 𝑏]. The fact that (𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]), ‖ ⋅ ‖𝑅𝐵𝑉𝑝 ) is continuously imbedded into both (𝐶([𝑎, 𝑏]), ‖ ⋅ ‖∞ ) and (𝐵𝑉([𝑎, 𝑏]), ‖ ⋅ ‖𝐵𝑉 ) follows from Proposition 1.3 (d) and (2.89). To prove the completeness of (𝑅𝐵𝑉 𝑝 ([𝑎, 𝑏]), ‖⋅‖𝑅𝐵𝑉𝑝 ), let (𝑓𝑛 )𝑛 be a Cauchy sequence in the norm (2.90). Given 𝜀 > 0, choose 𝑛0 ∈ ℕ such that 𝑚, 𝑛 ≥ 𝑛0 implies 𝑘

∑

|(𝑓𝑚 − 𝑓𝑛 )(𝑡𝑗 ) − (𝑓𝑚 − 𝑓𝑛 )(𝑡𝑗−1 )|𝑝 (𝑡𝑗 − 𝑡𝑗−1 )𝑝−1

𝑗=1

≤ Var𝑅𝑝 (𝑓𝑚 − 𝑓𝑛 , 𝑃) ≤ 𝜀𝑝

(2.92)

for every partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑘 } ∈ P([𝑎, 𝑏]). From what we have observed above, it follows that (𝑓𝑛 )𝑛 is a Cauchy sequence in the norm (0.45) of 𝐶([𝑎, 𝑏]). Thus, 𝑓𝑛 → 𝑓 uniformly on [𝑎, 𝑏], where 𝑓 is continuous on [𝑎, 𝑏]. Letting 𝑚 → ∞ in (2.92), we obtain 𝑘

∑ 𝑗=1

|(𝑓 − 𝑓𝑛 )(𝑡𝑗 ) − (𝑓 − 𝑓𝑛 )(𝑡𝑗−1 )|𝑝 (𝑡𝑗 − 𝑡𝑗−1 )𝑝−1

≤ 𝜀𝑝 ,

15 Here, the letter 𝑅 stands of course for “Riesz,” to distinguish this space from the space introduced in Definition 1.31.

2.4 The Riesz–Medvedev variation

| 163

and so Var𝑅𝑝 (𝑓 − 𝑓𝑛 )1/𝑝 ≤ 𝜀 for 𝑛 ≥ 𝑛0 . Consequently, 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) and ‖𝑓𝑛 − 𝑓‖𝑅𝐵𝑉𝑝 → 0 (𝑛 → ∞) . Formula (2.91) is proved in the same way as formula (1.17) in Proposition 1.10. The last assertion is obvious. A comparison with Definition 1.1 shows that Var𝑅1 (𝑓, 𝑃; [𝑎, 𝑏]) = Var(𝑓, 𝑃; [𝑎, 𝑏]),

Var𝑅1 (𝑓; [𝑎, 𝑏]) = Var(𝑓; [𝑎, 𝑏]) ,

and so 𝑅𝐵𝑉1 = 𝐵𝑉. Consequently, both spaces 𝑊𝐵𝑉𝑝 (Definition 1.31) and 𝑅𝐵𝑉𝑝 gen­ eralize the classical space 𝐵𝑉 in different directions. However, Definition 2.50 seems more interesting than Definition 1.31 since the space 𝑅𝐵𝑉𝑝 has two remarkable proper­ ties. Firstly, it has a natural characterization by means of absolutely continuous func­ tions (see Theorem 3.34 in the next chapter); secondly, it can be identified¹⁶ in a natural way with the dual space of the Lebesgue space 𝐿𝑝 ([𝑎, 𝑏]), as we will show in Section 4.2. In contrast to the space 𝐵𝑉([𝑎, 𝑏]) (or the spaces 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏])), all functions 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) are continuous in case 𝑝 > 1. This implies, in particular, that 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) is contained, for 𝑝 > 1, in the set 𝐶𝐵𝑉([𝑎, 𝑏]) introduced in Section 1.2. The ques­ tion arises if the spaces 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) are connected to the Lipschitz space 𝐿𝑖𝑝([𝑎, 𝑏]) or the Hölder spaces 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]), in a similar way as the spaces 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) (Proposi­ tion 1.34). The following result is parallel to the chain of inclusions (1.46); in particular, it shows that 𝑅𝐵𝑉𝑝 is intermediate between 𝐿𝑖𝑝 and 𝐴𝐶. Proposition 2.52. The inclusions 𝐿𝑖𝑝([𝑎, 𝑏]) ⊆ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊆ 𝐴𝐶([𝑎, 𝑏]) ⊆ 𝐵𝑉([𝑎, 𝑏])

(2.93)

hold for 𝑝 > 1. Proof. The proof of the first inclusion follows immediately from the estimate 𝑚

∑ 𝑗=1

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|𝑝 (𝑡𝑗 − 𝑡𝑗−1 )𝑝−1

𝑚

≤ 𝐿𝑝 ∑ (𝑡𝑗 − 𝑡𝑗−1 )𝑝−(𝑝−1) 𝑗=1

𝑝

𝑚

(2.94) 𝑝

= 𝐿 ∑ (𝑡𝑗 − 𝑡𝑗−1 ) = 𝐿 (𝑏 − 𝑎) 𝑗=1

for any function 𝑓 which satisfies (0.66). To prove the second inclusion, fix 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]), 𝑓(𝑥) ≢ 0, and let {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏]) be a finite collection of pairwise nonoverlapping subintervals of [𝑎, 𝑏]. Then 𝑛

𝑛

󸀠 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| (𝑏𝑘 − 𝑎𝑘 )1/𝑝 , 󸀠 1/𝑝 𝑘=1 (𝑏𝑘 − 𝑎𝑘 )

∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| = ∑ 𝑘=1

16 To be precise, to get the mentioned duality, we have to consider the subspace of all 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) satisfying 𝑓(𝑎) = 0, see Theorem 4.35.

164 | 2 Nonclassical BV-spaces where 𝑝󸀠 := 𝑝/(𝑝 − 1) as usual. Now, applying Hölder’s inequality (0.108) to 𝛼𝑘 :=

|𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| , (𝑏𝑘 − 𝑎𝑘 )1−1/𝑝

𝛽𝑘 := (𝑏𝑘 − 𝑎𝑘 )1/𝑝

󸀠

yields 𝑛

󸀠 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| (𝑏𝑘 − 𝑎𝑘 )1/𝑝 󸀠 1/𝑝 𝑘=1 (𝑏𝑘 − 𝑎𝑘 )

∑

1/𝑝

𝑛

|𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|𝑝 ≤ [∑ ] (𝑏𝑘 − 𝑎𝑘 )𝑝−1 𝑘=1 ≤

Var𝑅𝑝 (𝑓; [𝑎, 𝑏])1/𝑝

[ ∑ (𝑏𝑘 − 𝑎𝑘 )] 𝑘=1

𝑛

[ ∑ (𝑏𝑘 − 𝑎𝑘 )] 𝑘=1

1/𝑝󸀠

𝑛

1/𝑝󸀠

𝑛

1/𝑝󸀠

≤ ‖𝑓‖𝑅𝐵𝑉𝑝 [ ∑ (𝑏𝑘 − 𝑎𝑘 )]

.

𝑘=1

󸀠

𝑝󸀠

Therefore, if, for given 𝜀 > 0, we choose 𝛿 ≤ 𝜀𝑝 /‖𝑓‖𝑅𝐵𝑉𝑝 , then (1.40) implies (1.41), which means that 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) as claimed. The fact that every absolutely continuous function has bounded variation has al­ ready been proved in Proposition 1.22. In Theorem 3.34 in the next chapter, we will give a precise characterization of those functions 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]), in terms of their derivatives a.e. which belong to 𝑅𝐵𝑉 𝑝 ([𝑎, 𝑏]). Observe that (2.90) and (2.94) show that ‖𝑓‖𝑅𝐵𝑉𝑝 = |𝑓(𝑎)| + Var𝑅𝑝 (𝑓; [𝑎, 𝑏])1/𝑝 ≤ |𝑓(𝑎)| + 𝐿 if 𝑓 satisfies (0.66). Taking 𝐿 as close as we want to the minimal Lipschitz con­ stant (0.68) of 𝑓, we conclude that (𝐿𝑖𝑝([𝑎, 𝑏]), ‖ ⋅ ‖𝐿𝑖𝑝 ) is continuously imbedded into (𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]), ‖ ⋅ ‖𝑅𝐵𝑉𝑝 ). Similarly, the norm estimate ‖𝑓‖𝐵𝑉 ≤ ‖𝑓‖𝑅𝐵𝑉𝑝 for all 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) means that 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) 󳨅→ 𝐵𝑉([𝑎, 𝑏]) with 𝑐(𝑅𝐵𝑉𝑝 , 𝐵𝑉) = 1. More­ over, letting 𝑝 → ∞ in (2.87), we see that we may identify the space 𝑅𝐵𝑉∞ ([𝑎, 𝑏]) formally with the space 𝐿𝑖𝑝([𝑎, 𝑏]). In this sense, we may consider (2.93) as “interpo­ lation inclusion” 𝑅𝐵𝑉∞ ([𝑎, 𝑏]) ⊂ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊂ 𝑅𝐵𝑉1 ([𝑎, 𝑏])

(1 < 𝑝 < ∞) .

(2.95)

Since 𝑅𝐵𝑉𝑝 ⊆ 𝐴𝐶 for 𝑝 > 1, see (2.93), one cannot expect that the space 𝑅𝐵𝑉𝑝 contains some Hölder space¹⁷ 𝐿𝑖𝑝𝛼 for a suitable choice of 𝛼 < 1. The next example shows this in explicit form. Example 2.53. Let 𝑓 : [0, 1] → ℝ be defined as in Example 1.24. Then 𝑓 belongs to each Hölder space 𝐿𝑖𝑝𝛼 ([0, 1]) for 𝛼 < 1. However, 𝑓 cannot belong to 𝑅𝐵𝑉𝑝 ([0, 1]) for any choice of 𝑝 ≥ 1, by (2.93), since 𝑓 ∈ ̸ 𝐵𝑉([0, 1]). ♥

17 So, an analogue to Proposition 1.34 is not true for 𝑊𝐵𝑉𝑝 replaced by 𝑅𝐵𝑉𝑝 .

2.4 The Riesz–Medvedev variation

| 165

One could also ask if a parallel result as Proposition 1.38 holds for the spaces 𝑅𝐵𝑉𝑝 , i.e. whether or not the space 𝑅𝐵𝑉𝑝 is contained in some space 𝑅𝐵𝑉𝑞 for suitable values of 𝑝 and 𝑞. In fact, the following is true. Proposition 2.54. Let 1 ≤ 𝑝 ≤ 𝑞 < ∞. Then the inclusion 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊇ 𝑅𝐵𝑉𝑞 ([𝑎, 𝑏]) holds. Proof. Without loss of generality, let 1 < 𝑝 < 𝑞; then, 𝑞 > 1, 𝑝

𝑟 :=

𝑠 :=

𝑞 > 1, 𝑞−𝑝

1 1 + = 1. 𝑟 𝑠

Consequently, given 𝑓 ∈ 𝑅𝐵𝑉𝑞 ([𝑎, 𝑏]) and any partition 𝑃 := {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), from Hölder’s inequality (0.108) for 𝛼𝑗 :=

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|𝑝 (𝑡𝑗 − 𝑡𝑗−1 )𝑝−1+1/𝑠

𝛽𝑗 := (𝑡𝑗 − 𝑡𝑗−1 )1/𝑠

,

(and (𝑝, 𝑝󸀠 ) replaced by (𝑟, 𝑠)), we get 𝑚

∑ 𝑗=1

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|𝑝 (𝑡𝑗 − 𝑡𝑗−1 )𝑝−1 |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|𝑝

𝑚

=∑ 𝑗=1

(𝑡𝑗 − 𝑡𝑗−1 )𝑝−1+1/𝑠

𝑚

= (∑ 𝑗=1

(𝑡𝑗 − 𝑡𝑗−1 )

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| (𝑡𝑗 − 𝑡𝑗−1 )𝑞−1

1/𝑠

𝑚

≤ (∑ 𝑗=1

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|𝑝𝑟 (𝑡𝑗 − 𝑡𝑗−1 )(𝑝−1+1/𝑠)𝑟

1/𝑟

)

1/𝑠

𝑚

(∑ (𝑡𝑗 − 𝑡𝑗−1 )) 𝑗=1

𝑝/𝑞

𝑞

)

(𝑏 − 𝑎)(𝑞−𝑝)/𝑞 ≤ Var𝑅𝑞 (𝑓; [𝑎, 𝑏])𝑝/𝑞 (𝑏 − 𝑎)(𝑞−𝑝)/𝑞 ,

and so 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) as claimed. Another method for proving Proposition 2.54 will be given in Theorem 3.34 in the next chapter. As the inclusion in Proposition 1.38, the inclusion in Proposition 2.54 is strict in case 𝑝 < 𝑞, see Exercise 2.26. In spite of their similarity, the Wiener spaces 𝑊𝐵𝑉𝑝 introduced in Definition 1.31 and the Riesz spaces 𝑅𝐵𝑉𝑝 introduced in Definition 2.50 have quite different properties for 𝑝 > 1. We compare some of them in Table 2.3 below. We now come to a new function class which has been introduced by Medvedev [211] and which generalizes the space 𝑅𝐵𝑉𝑝 in rather the same way as the class 𝑊𝐵𝑉𝜙 generalizes the space 𝑊𝐵𝑉𝑝 , see Definition 2.2. Definition 2.55. Given a Young function 𝜙, a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), and a function 𝑓 : [𝑎, 𝑏] → ℝ, the nonnegative real number 𝑚

Var𝑅𝜙 (𝑓, 𝑃) = Var𝑅𝜙 (𝑓, 𝑃; [𝑎, 𝑏]) := ∑ 𝜙 ( 𝑗=1

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| 𝑡𝑗 − 𝑡𝑗−1

) (𝑡𝑗 − 𝑡𝑗−1 )

(2.96)

166 | 2 Nonclassical BV-spaces Table 2.3. 𝑊𝐵𝑉𝑝 vs. 𝑅𝐵𝑉𝑝 (1 < 𝑝 < ∞). Space

𝑊𝐵𝑉𝑝 ([𝑎, 𝑏])

𝑅𝐵𝑉𝑝 ([𝑎, 𝑏])

definition dependence on 𝑝 for 𝑝 = 1 coincides with for 𝑝 = ∞ coincides with is contained in 𝐶([𝑎, 𝑏]) is contained in 𝐵𝑉([𝑎, 𝑏]) contains 𝐿𝑖𝑝([𝑎, 𝑏]) contains 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏])

Wiener (1924) increasing 𝐵𝑉([𝑎, 𝑏]) 𝑅([𝑎, 𝑏]) no no yes yes (for 𝛼 ≤ 1/𝑝)

Riesz (1910) decreasing 𝐵𝑉([𝑎, 𝑏]) 𝐿𝑖𝑝([𝑎, 𝑏]) yes yes yes no

is called the Riesz–Medvedev variation of 𝑓 on [𝑎, 𝑏] with respect to 𝑃, while the (pos­ sibly infinite) number Var𝑅𝜙 (𝑓) = Var𝑅𝜙 (𝑓; [𝑎, 𝑏]) := sup {Var𝑅𝜙 (𝑓, 𝑃; [𝑎, 𝑏]) : 𝑃 ∈ P([𝑎, 𝑏])} ,

(2.97)

where the supremum is taken over all partitions of [𝑎, 𝑏], is called the total Riesz–Medvedev variation of 𝑓 on [𝑎, 𝑏]. In case Var𝑅𝜙 (𝑓; [𝑎, 𝑏]) < ∞, we say that 𝑓 has bounded Riesz–Medvedev variation (or bounded 𝜙-variation in Riesz’s sense) on [𝑎, 𝑏] and write ◼ 𝑓 ∈ 𝑉𝜙𝑅 ([𝑎, 𝑏]). Similarly as in (2.10) and (2.11), we put 𝐵𝑅 (𝜙) := {𝑓 ∈ 𝐵([𝑎, 𝑏]) : Var𝑅𝜙 (𝑓; [𝑎, 𝑏]) ≤ 1} ,

(2.98)

‖𝑓‖𝑅𝐵𝑉𝜙 := |𝑓(𝑎)| + inf {𝜆 > 0 : 𝑓/𝜆 ∈ 𝐵𝑅 (𝜙)} ,

(2.99)

and¹⁸ and denote its linear hull span 𝐵𝑅 (𝜙) by 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏])). Proposition 2.56. The inclusions 𝐿𝑖𝑝([𝑎, 𝑏]) ⊆ 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) ⊆ 𝐵𝑉([𝑎, 𝑏])

(2.100)

are true. Proof. Let 𝑓 ∈ 𝐿𝑖𝑝([𝑎, 𝑏]). From |𝑓(𝑠) − 𝑓(𝑡)| ≤ 𝐿|𝑠 − 𝑡|, it then follows that 𝑚

∑𝜙( 𝑗=1

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| 𝑡𝑗 − 𝑡𝑗−1

𝑚

) (𝑡𝑗 − 𝑡𝑗−1 ) ≤ ∑ 𝜙(𝐿)(𝑡𝑗 − 𝑡𝑗−1 ) = 𝜙(𝐿)(𝑏 − 𝑎) , 𝑗=1

which shows that 𝑓 ∈ 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) with ‖𝑓‖𝑅𝐵𝑉𝜙 ≤ |𝑓(𝑎)| + 𝜙(𝑙𝑖𝑝(𝑓))(𝑏 − 𝑎) ,

18 Clearly, for 𝜙(𝑡) = 𝑡𝑝 with 𝑝 ≥ 1, the infimum in (2.99) coincides with Var𝑅𝑝 (𝑓; [𝑎, 𝑏])1/𝑝 , see (2.88).

2.4 The Riesz–Medvedev variation

|

167

where 𝑙𝑖𝑝(𝑓) denotes the smallest Lipschitz constant (0.68) of 𝑓. To prove the second inclusion in (2.100), assume that 𝑓 ∈ 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]). Then 𝑚

𝑚

∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| = ∑

𝑗=1

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗 − 1)| 𝑡𝑗 − 𝑡𝑗−1

𝑗=1

𝑚

= 𝜙−1 (𝜙 [ ∑ [𝑗=1

(𝑡𝑗 − 𝑡𝑗−1 )

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|

𝑚

𝑡𝑗 − 𝑡𝑗−1

= 𝜙−1 (𝜙 [ ∑ (𝑏 − 𝑎) [𝑗=1

(𝑡𝑗 − 𝑡𝑗−1 )]) ]

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| 𝑡𝑗 − 𝑡𝑗−1 𝑡𝑗 − 𝑡𝑗−1

𝑏−𝑎

]) . ]

Hence, using the fact that 𝜙−1 is increasing, we obtain 𝑚

𝑚

∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| ≤ 𝜙−1 (∑ 𝜙 [(𝑏 − 𝑎)

𝑗=1

𝑗=1

≤ 𝜙−1 ( ≤ 𝜙−1 (

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| 𝑡𝑗 − 𝑡𝑗−1

]

𝑡𝑗 − 𝑡𝑗−1 𝑏−𝑎

)

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| 1 𝑚 ] (𝑡𝑗 − 𝑡𝑗−1 )) ∑ 𝜙 [(𝑏 − 𝑎) 𝑏 − 𝑎 𝑗=1 𝑡𝑗 − 𝑡𝑗−1

1 Var𝑅𝜙 ((𝑏 − 𝑎)𝑓; [𝑎, 𝑏])) < ∞ . 𝑏−𝑎

This shows that 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) and proves the second inclusion in (2.100). Let us return for a moment to the condition ∞𝑝 introduced in Definition 2.11. This condition was important in the discussion of the Wiener–Young space 𝑊𝐵𝑉𝜙 . It turns out that it is also useful in the discussion of the Riesz–Medvedev space 𝑅𝐵𝑉𝜙 . In fact, one can show that if the second inclusion in (2.100) is strict, then the Young function 𝜙 necessarily satisfies condition ∞1 . To see this, suppose that 𝜙 ∈ ̸ ∞1 which means that lim

𝑡→∞

𝜙(𝑡) < ∞, 𝑡

(2.101)

see (2.16). Then there exist constants 𝑇 > 0 and 𝑀 > 0 such that 𝜙(𝑡) ≤ 𝑀𝑡 for 𝑡 ≥ 𝑇. Given an arbitrary partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), we define an index set 𝐴(𝑇) ⊆ {1, 2, . . . , 𝑚} by 𝐴(𝑇) := {𝑗 ∈ {1, 2, . . . , 𝑚}, |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| ≥ 𝑇(𝑡𝑗 − 𝑡𝑗−1 )}

168 | 2 Nonclassical BV-spaces and put 𝐵(𝑇) := {1, 2, . . . , 𝑚} \ 𝐴(𝑇). Then we have 𝑚

Var𝑅𝜙 (𝑓, 𝑃 : [𝑎, 𝑏]) = ∑ 𝜙 (

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| 𝑡𝑗 − 𝑡𝑗−1

𝑗=1

= ∑ 𝜙(

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|

𝑗∈𝐴(𝑇)

+ ∑ 𝜙( 𝑗∈𝐵(𝑇)

≤ ∑ 𝑀 𝑗∈𝐴(𝑇)

) (𝑡𝑗 − 𝑡𝑗−1 )

𝑡𝑗 − 𝑡𝑗−1

) (𝑡𝑗 − 𝑡𝑗−1 )

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| 𝑡𝑗 − 𝑡𝑗−1

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| 𝑡𝑗 − 𝑡𝑗−1

) (𝑡𝑗 − 𝑡𝑗−1 )

(𝑡𝑗 − 𝑡𝑗−1 ) + ∑ 𝜙(𝑇)(𝑡𝑗 − 𝑡𝑗−1 ) 𝑗∈𝐵(𝑇)

≤ 𝑀 Var(𝑓, 𝑃 : [𝑎, 𝑏]) + 𝜙(𝑇)(𝑏 − 𝑎) ≤ 𝑀 Var(𝑓; [𝑎, 𝑏]) + 𝜙(𝑇)(𝑏 − 𝑎) . Passing to the supremum with respect to 𝑃 ∈ P([𝑎, 𝑏]), we conclude that Var𝑅𝜙 (𝑓; [𝑎, 𝑏]) ≤ 𝑀 Var(𝑓; [𝑎, 𝑏]) + 𝜙(𝑇)(𝑏 − 𝑎) , and so 𝐵𝑉([𝑎, 𝑏]) ⊆ 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]). The converse inclusion has already been proved in Proposition 2.56 for arbitrary Young functions 𝜙. We summarize our discussion with the following Proposition 2.57. Let 𝜙 be a Young function which does not satisfy the condition ∞1 . Then the space 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) coincides with 𝐵𝑉([𝑎, 𝑏]). We do not know whether or not the following generalization of Proposition 2.57 holds true: if 𝜙 is a Young function which does not satisfy the condition ∞𝑝 , see (2.16), then the space 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) coincides with the space 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) introduced in Defi­ nition 2.50. To conclude this section, let us again consider our favorite families of functions which we introduced in the last part of Section 0.3, namely, the oscillatory functions (0.86) and the special zigzag functions (0.93). It is illuminating to determine all val­ ues of 𝛼, 𝛽 ∈ ℝ for which the function (0.86) belongs to one of the function classes considered so far, and also all values of 𝜃 ∈ ℝ for which the function (0.93) belongs to such classes. In the synoptic Table 2.4 below, which essentially extends Table 0.1, we do this for the classes 𝐶, 𝐶1 , 𝐿 1 , 𝐿𝑖𝑝, 𝐿𝑖𝑝𝛾 (0 < 𝛾 < 1), 𝐵𝑉, 𝑊𝐵𝑉𝑝 (1 < 𝑝 < ∞), 𝑅𝐵𝑉𝑝 (1 < 𝑝 < ∞), and 𝐴𝐶. Let us explain the new entries in Table 2.4. The admissible values for 𝛼 and 𝛽 for the function (0.86) have already been collected in Proposition 0.48 and Exercises 0.52, 1.8, 1.9, 1.57, and 2.27. From (0.17) and (1.75), it follows that 𝑍𝜃 ∈ 𝑊𝐵𝑉𝑝 ([0, 1]) if and only if the series ∞ 1 𝜁(𝑘𝜃, 0) = ∑ 𝑝𝜃 𝑘 𝑘=1

2.5 The Korenblum variation

| 169

Table 2.4. Oscillation functions and zigzag functions.

belongs to 𝐶([0, 1]) iff belongs to 𝐶1 ([0, 1]) iff belongs to 𝐿 1 ([0, 1]) iff belongs to 𝐿𝑖𝑝([0, 1]) iff belongs to 𝐿𝑖𝑝𝛾 ([0, 1]) iff belongs to 𝐵𝑉([0, 1]) iff belongs to 𝑊𝐵𝑉𝑝 ([0, 1]) iff belongs to 𝑅𝐵𝑉𝑝 ([0, 1]) iff belongs to 𝐴𝐶([0, 1]) iff

The function 𝑓𝛼,𝛽

The function 𝑍𝜃

𝛼 > 0 or 𝛼 ≤ 0 and 𝛼 + 𝛽 > 0 𝛼+𝛽 >1 𝛼+𝛽 ≥1 𝛼+𝛽 ≥1 see Exercise 0.52 𝛽 > 0 and 𝛼 + 𝛽 ≥ 0 or 𝛽 ≤ 0 and 𝛼 + 𝛽 > 0 𝛽 > 0 and 𝑝𝛼 + 𝛽 ≥ 0 or 𝛽 ≤ 0 and 𝑝𝛼 + 𝛽 > 0 see Exercise 2.27 𝛼+𝛽 >0

always never always never never 𝜃>1 𝑝𝜃 > 1 𝑝 = 1 and 𝜃 > 1 𝜃>1

converges, i.e. precisely for 𝑝𝜃 > 1; in particular, 𝑍𝜃 ∈ 𝐵𝑉([0, 1]) if and only if 𝜃 > 1. A similar computation shows that ∞

2𝑘(𝑝−1) . 𝑝𝜃 𝑘=1 𝑘

Var𝑅𝑝 (𝑍𝜃 ; [0, 1]) = ∑

(2.102)

Since 𝑝 ≥ 1, elementary convergence criteria show that 𝑍𝜃 ∈ 𝑅𝐵𝑉𝑝 ([0, 1]) if 𝑝 = 1 and 𝜃 > 1. Finally, we have already shown in Corollary 0.51 that 𝑍𝜃 ∈ ̸ 𝐿𝑖𝑝𝛾 ([0, 1]) for any 𝜃 > 0 and 𝛾 ∈ (0, 1]. Table 2.4 shows that the zigzag function (0.93) exhibits a more interesting behavior in spaces of (generalized) bounded variation than in the other spaces occurring in Table 0.1.

2.5 The Korenblum variation As we have seen in this and the preceding chapter, the concept of variation has been generalized in many directions. Wiener [321] distorted the measurement of intervals in the range using powers |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|𝑝 , see Definition 1.31, while Young [322] used more general distortions of the form 𝜙(|𝑓(𝑡𝑗 )−𝑓(𝑡𝑗−1 )|), see Definition 2.2. The most general concept by Schramm [286], see Definition 2.42, replaced such distortion functions 𝜙 by countable families 𝛷. All of these extensions have the advantage of allowing us to define quite general Riemann–Stieltjes integrals, as we shall see in Chapter 4. On the other hand, a flaw is the loss of an effective decomposition of a function from the corresponding classes 𝑊𝐵𝑉𝑝 , 𝑊𝐵𝑉𝜙 , and 𝛷𝐵𝑉 into, hopefully, simpler functions, such as in Theorem 1.5 for the case of functions from the classical space 𝐵𝑉.

170 | 2 Nonclassical BV-spaces In 1975, Korenblum [163] considered a new kind of variation which he called 𝜅variation, introducing a function 𝜅 for distorting the expression |𝑡𝑗 − 𝑡𝑗−1 | in the par­ tition itself, rather than the expression |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| in the range. Subsequently, this class of functions has been studied in detail by Cyphert and Kalingos [102]. One advantage of this alternate approach is that a function of bounded 𝜅-variation may be decomposed into the difference of two simpler functions called 𝜅-decreasing functions (for the precise definitions, see below). We will follow the article [102] in the sequel and will mostly consider, without loss of generality, functions over [𝑎, 𝑏] = [0, 1]. Definition 2.58. A function 𝜅 : [0, 1] → [0, 1] is called a distortion function if 𝜅 is increasing, concave, and satisfies 𝜅(0) = 0, 𝜅(1) = 1, and lim

𝑡→0+

𝜅(𝑡) = ∞, 𝑡

(2.103)

i.e. has infinite slope at the origin.

◼

Note that from the estimate 𝜅(𝑠 + 𝑡) − 𝜅(𝑡) 𝜅(𝑠) − 𝜅(0) ≤ , (𝑠 + 𝑡) − 𝑡 𝑠−0 it follows that a distortion function is always subadditive in the sense that 𝜅(𝑠 + 𝑡) ≤ 𝜅(𝑠) + 𝜅(𝑡)

(0 ≤ 𝑠, 𝑡 ≤ 1) .

(2.104)

Moreover, the definition implies that a distortion function is continuous¹⁹ on (0, 1]. Here are some typical examples of distortion functions. Example 2.59. The simplest example is of course 𝜅(𝑡) = 𝑡𝛼

(0 < 𝛼 < 1) .

Since 𝜅󸀠 (𝑡) = 𝛼𝑡𝛼−1 > 0 and 𝜅󸀠󸀠 (𝑡) = 𝛼(𝛼 − 1)𝑡𝛼−2 < 0 on (0, 1), 𝜅 is increasing and concave. Another example is {𝑡(1 − log 𝑡) 𝜅(𝑡) = { 0 {

for 0 < 𝑡 ≤ 1 , for 𝑡 = 0 ;

here, we have 𝜅󸀠 (𝑡) = − log 𝑡 > 0 and 𝜅󸀠󸀠 (𝑡) = − 1𝑡 < 0 on (0, 1). Finally, the function { 2 for 0 < 𝑡 ≤ 1 , 𝜅(𝑡) = { 2−log 𝑡 0 for 𝑡 = 0 { is a distortion function which arises in entropy theory, see Section 2.8.

♥

19 A comparison with Definition 0.52 shows that distortion functions are similar to moduli of conti­ nuity; in particular, they are both increasing and subadditive. However, the main difference is that a modulus is continuous at zero, while a distortion function is continuous everywhere except, perhaps, at zero.

2.5 The Korenblum variation

|

171

Building on the concept of distortion functions, we may now introduce a new class of functions of bounded variation. Definition 2.60. Given a distortion function 𝜅 : [0, 1] → [0, 1], a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]), and a function 𝑓 : [0, 1] → ℝ, the nonnegative real number Var𝜅 (𝑓, 𝑃) = Var𝜅 (𝑓, 𝑃; [0, 1]) :=

∑𝑚 𝑗=1 |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|

(2.105)

∑𝑚 𝑗=1 𝜅(𝑡𝑗 − 𝑡𝑗−1 )

is called the Korenblum variation of 𝑓 on [0, 1] with respect to 𝑃, while the (possibly infinite) number Var𝜅 (𝑓) = Var𝜅 (𝑓; [0, 1]) := sup {Var𝜅 (𝑓, 𝑃; [0, 1]) : 𝑃 ∈ P([0, 1])} ,

(2.106)

where the supremum is taken over all partitions of [0, 1], is called the total Korenblum variation of 𝑓 on [0, 1]. In case Var𝜅 (𝑓; [0, 1]) < ∞, we say that 𝑓 has bounded Koren­ blum variation (or bounded 𝜅-variation) on [0, 1] and write 𝑓 ∈ 𝜅𝐵𝑉([0, 1]). ◼ Of course, Definition 2.60 may also be formulated for functions on an arbitrary interval [𝑎, 𝑏] by defining that 𝑓 belongs to 𝜅𝐵𝑉([𝑎, 𝑏]) if the function 𝑥 󳨃→ 𝑓((𝑏−𝑎)𝑥+𝑎) belongs to 𝜅𝐵𝑉([0, 1]). This amounts to replacing (2.105) by Var𝜅 (𝑓, 𝑃) = Var𝜅 (𝑓, 𝑃; [0, 1]) :=

∑𝑚 𝑗=1 |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| ∑𝑚 𝑗=1 𝜅 (

𝑡𝑗 −𝑡𝑗−1 ) 𝑏−𝑎

.

(2.107)

From (2.104), it then follows that 𝑚

∑𝜅(

𝑡𝑗 − 𝑡𝑗−1 𝑏−𝑎

𝑗=1

𝑚

) ≥ 𝜅 (∑ 𝑗=1

𝑡𝑗 − 𝑡𝑗−1 𝑏−𝑎

) = 𝜅(1) = 1 .

In the following proposition, we compare 𝜅𝐵𝑉([0, 1]) with other function classes. Proposition 2.61. For any distortion function 𝜅, the inclusion 𝐵𝑉([0, 1]) ⊆ 𝜅𝐵𝑉([0, 1]) ⊆ 𝑅([0, 1])

(2.108)

holds, where 𝑅([𝑎, 𝑏]) denotes the set of regular functions introduced in Section 0.3. Proof. By (2.104), we have 𝑚

𝑚

∑ 𝜅(𝑡𝑗 − 𝑡𝑗−1 ) ≥ 𝜅 (∑ (𝑡𝑗 − 𝑡𝑗−1 )) = 𝜅(1) = 1 ,

𝑗=1

and hence

𝑗=1

𝑚

Var𝜅 (𝑓, 𝑃; [0, 1]) ≤ ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| = Var(𝑓, 𝑃; [0, 1]) 𝑗=1

172 | 2 Nonclassical BV-spaces for every partition 𝑃 ∈ P([0, 1]). This proves the first inclusion in (2.108). To prove the second inclusion, we have to show that every function 𝑓 ∈ 𝜅𝐵𝑉([0, 1]) is bounded and has unilateral limits at each point 𝑥0 ∈ [0, 1]. By considering the partition 𝑃𝑥 := {0, 𝑥, 1}, we see that |𝑓(𝑥) − 𝑓(0)| ≤

3 3 Var𝜅 (𝑓, 𝑃𝑥 ; [0, 1]) ≤ Var𝜅 (𝑓; [0, 1]) 2 2

(0 ≤ 𝑥 ≤ 1) ,

and so

3 Var𝜅 (𝑓) , (2.109) 2 where ‖ ⋅ ‖∞ denotes the norm (0.39). To show the existence of the right limit at 𝑥0 ∈ [0, 1), say, suppose that ‖𝑓‖∞ ≤ |𝑓(0)| +

𝑙 := lim inf 𝑓(𝑥) < lim sup 𝑓(𝑥) =: 𝐿 . 𝑥→𝑥0 +

𝑥→𝑥0 +

Then for each sufficiently large 𝑛 ∈ ℕ, one can choose points 𝑡1 , 𝑡2 , . . . , 𝑡𝑛 , 𝑡𝑛+1 such that 𝑥0 < 𝑡1 < 𝑡2 < . . . < 𝑡𝑛 < 𝑡𝑛+1 ≤ 𝑥0 +

1 𝑛

and

𝐿−𝑙 (𝑗 = 1, 2, . . . , 𝑛) . 2 Using the partition 𝑃 = {0, 𝑡1 , 𝑡2 , . . . , 𝑡𝑛 , 𝑡𝑛+1 , 1} ∈ P([0, 1]) and the definition (2.105), we obtain |𝑓(𝑡𝑗+1 ) − 𝑓(𝑡𝑗 )| ≥

𝑛 𝑛(𝐿 − 𝑙) ≤ |𝑓(𝑡1 ) − 𝑓(0)| + ∑ |𝑓(𝑡𝑗+1 ) − 𝑓(𝑡𝑗 )| + |𝑓(1) − 𝑓(𝑡𝑛+1 )| 2 𝑗=1 𝑛

≤ Var𝜅 (𝑓, 𝑃)[𝜅(𝑡1 ) + ∑ 𝜅(𝑡𝑗+1 − 𝑡𝑗 ) + 𝜅(1 − 𝑡𝑛+1 )] ≤ Var𝜅 (𝑓, 𝑃) [𝑛𝜅(1/𝑛) + 2] . 𝑗=1

Dividing by 𝑛 and letting 𝑛 → ∞ yields 𝐿 − 𝑙 = 0, that is, a contradiction. The existence of the left-hand limit at any 𝑥0 ∈ (0, 1] is proved in the same way. We point out that the slope condition (2.104) ensures that the first inclusion in (2.108) is strict. In fact, if the limit in (2.103) is finite, one may show that 𝜅𝐵𝑉([0, 1]) = 𝐵𝑉([0, 1]). To get an idea of how to calculate the Korenblum variation, we consider a very simple class of functions. Example 2.62. Suppose that 𝑓 : [0, 1] → ℝ only takes two values, say {𝐴 for 0 ≤ 𝑥 ≤ 𝜉 , 𝑓(𝑥) := { 𝐵 for 𝜉 < 𝑥 ≤ 1 { for some fixed 𝜉 ∈ (0, 1). For each partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝜉, . . . , 𝑡𝑚 } ∈ P([0, 1]), we obtain, by (2.104), Var𝜅 (𝑓, 𝑃; [0, 1]) =

∑𝑚 𝑗=1 |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| ∑𝑚 𝑗=1 𝜅(𝑡𝑗 − 𝑡𝑗−1 )

≤ |𝐴 − 𝐵| ,

2.5 The Korenblum variation

| 173

and hence Var𝜅 (𝑓; [0, 1]) ≤ |𝐴 − 𝐵|. More generally, this implies that 𝜅𝐵𝑉 contains all step functions. ♥ At this point, one could ask whether or not the class 𝜅𝐵𝑉([0, 1]) is related to monotone functions as the class 𝐵𝑉([0, 1]). It is easy to see that every monotone function 𝑓 : [0, 1] → ℝ belongs to 𝜅𝐵𝑉([0, 1]) with Var𝜅 (𝑓; [0, 1]) ≤ |𝑓(1) − 𝑓(0)| .

(2.110)

This is completely analogous to Proposition 1.3 (e) and may be proved by using the trivial partition 𝑃 = {0, 1} ∈ P([0, 1]) which is optimal for (2.105) for monotone 𝑓. However, to formulate a decomposition theorem for functions of bounded Korenblum variation, one has to replace monotonicity by the following notion which has been studied in [CyK]. Definition 2.63. Given a distortion function 𝜅 : [0, 1] → [0, 1], a function 𝑓 : [0, 1] → ℝ is said to be 𝜅-decreasing if there exists a constant 𝐶 ≥ 0 such that 𝑓(𝑦) − 𝑓(𝑥) ≤ 𝐶𝜅(𝑦 − 𝑥)

(0 ≤ 𝑥 < 𝑦 ≤ 1) .

(2.111)

We write 𝐶𝜅 (𝑓) = 𝐶𝜅 (𝑓; [0, 1]) := sup {

𝑓(𝑦) − 𝑓(𝑥) : 0 ≤ 𝑥 < 𝑦 ≤ 1} 𝜅(𝑦 − 𝑥)

for the smallest 𝐶 ≥ 0 satisfying (2.111).

(2.112) ◼

Let us make some remarks about this definition. Clearly, every decreasing function 𝑓 is 𝜅-decreasing, for any distortion function 𝜅, with 𝐶𝜅 (𝑓) = 0. However, the class of 𝜅-decreasing functions is much larger. To see this, observe that any function 𝑓 ∈ 𝐿𝑖𝑝𝛼 ([0, 1]) is 𝜅-decreasing for 𝜅(𝑡) = 𝑡𝛼 (0 < 𝛼 < 1) and 𝐶𝜅 (𝑓) = 𝑙𝑖𝑝𝛼 (𝑓), see (0.69). This shows, in particular, that continuous nowhere differentiable functions which are 𝜅-decreasing exist. Intuitively, a function 𝑓 is 𝜅-decreasing if 𝑓 is either decreasing or, at least locally, not increasing faster than some fixed multiple of 𝜅 itself. The following proposition provides an important link between 𝜅-decreasing func­ tions and functions of bounded 𝜅-variation. Here and in what follows, we use the no­ tation 𝑎+ := max {𝑎, 0}, 𝑎− := max {−𝑎, 0} (𝑎 ∈ ℝ) ; (2.113) so, we have 𝑎+ + 𝑎− = |𝑎| and 𝑎+ − 𝑎− = 𝑎. Proposition 2.64. Every 𝜅-decreasing function 𝑓 : [0, 1] → ℝ has bounded 𝜅-variation with Var𝜅 (𝑓; [0, 1]) ≤ 2𝐶𝜅 (𝑓) + |𝑓(1) − 𝑓(0)| . (2.114) Proof. Suppose first that 𝑓(0) = 𝑓(1), and let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]) be fixed. From (2.112), it follows that 𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 ) ≤ 𝐶𝜅 (𝑓)𝜅(𝑡𝑗 − 𝑡𝑗−1 ) (𝑗 = 1, 2, . . . , 𝑚) .

174 | 2 Nonclassical BV-spaces So, the equality |𝑎| = 𝑎+ + 𝑎− implies that 𝑚

𝑚

𝑗=1

𝑗=1

𝑚

∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| = ∑ [𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )]+ + ∑ [𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )]− 𝑗=1

𝑚

≤ 2𝐶𝜅 (𝑓) ∑ 𝜅(𝑡𝑗 − 𝑡𝑗−1 ), 𝑗=1

showing that 𝑓 ∈ 𝜅𝐵𝑉([0, 1]) with Var𝜅 (𝑓; [0, 1]) ≤ 2𝐶𝜅 (𝑓). The proof in the general case is similar and left to the reader (Exercise 2.33). Observe that (2.114) generalizes (2.110) since 𝐶𝜅 (𝑓) = 0 for monotone functions 𝑓. Now, we present some examples which give some indication as to the flavor of the subject, and, in particular, of the abstract Definitions 2.60 and 2.63. The first example shows that not every function 𝑓 ∈ 𝜅𝐵𝑉([0, 1]) is 𝜅-decreasing. Example 2.65. Let 𝜅 : [0, 1] → [0, 1] be an arbitrary distortion function, and define 𝑓 : [0, 1] → ℝ by 𝑓(𝑥) := √𝜅(𝑥). Then 𝑓 is monotonically increasing, and so has bounded 𝜅-variation with Var𝜅 (𝑓; [0, 1]) = 1 by (2.110). On the other hand, lim

𝑥→0+

𝑓(𝑥) − 𝑓(0) 1 = ∞, = lim 𝑥→0+ √𝜅(𝑥) 𝜅(𝑥 − 0) ♥

and so 𝑓 cannot satisfy (2.111) for any 𝐶 > 0.

Let us return to the inclusions (2.108). To find an example of a regular function which does not belong to 𝜅𝐵𝑉([0, 1]), for a given distortion function, 𝜅 is easy (Exercise 2.34). In the next example, we also show that the first inclusion in (2.108) is strict by con­ structing a function 𝑓 ∈ 𝜅𝐵𝑉([0, 1]) \ 𝐵𝑉([0, 1]). Example 2.66. Let 𝜅 : [0, 1] → [0, 1] be defined by 𝜅(𝑡) = 𝑡𝛼 for some 𝛼 ∈ (0, 1). Using the notation (0.17), put 𝑡𝑛 :=

𝑛 1 1 ∑ 1/𝛼 𝜁(1/𝛼, 0) 𝑘=1 𝑘

(𝑛 = 1, 2, 3, . . .) .

We define a function 𝑓 : [0, 1] → ℝ by {0 𝑓(𝑥) := { (𝑥 − 𝑡𝑛 )𝛼 {

for 𝑥 = 0 or 𝑥 = 1 , for 𝑡𝑛 ≤ 𝑥 < 𝑡𝑛+1 .

Considering partitions containing the points 𝑡0 , 𝑡1 , . . . , 𝑡𝑛 , one easily sees that ∞

Var(𝑓; [0, 1]) ≥ 2 ∑ 𝜅 ( 𝑘=1

∞ 1 = ∞, ) = 2 ∑ 1/𝛼 𝑘 𝑘 𝑘=1

1

and so 𝑓 ∈ ̸ 𝐵𝑉([0, 1]). On the other hand, 𝑓 is 𝜅-decreasing. To see this, fix 𝑥 and 𝑦 with 0 ≤ 𝑥 < 𝑦 ≤ 1. Then there exists unique integers 𝑚, 𝑛 ∈ ℕ, 𝑚 ≤ 𝑛, such that

2.5 The Korenblum variation

| 175

𝑥 ∈ [𝑡𝑛 , 𝑡𝑛+1 ) and 𝑦 ∈ [𝑡𝑚 , 𝑡𝑚+1 ). Then 𝑡𝑛 ≤ 𝑡𝑚 , by construction, and the definition of 𝑓 implies that 𝑓(𝑦) − 𝑓(𝑥) = 𝜅(𝑦 − 𝑡𝑚 ) − 𝜅(𝑥 − 𝑡𝑛 ) ≤ 𝜅(𝑦 − 𝑡𝑛 ) − 𝜅(𝑥 − 𝑡𝑛 ) ≤ 𝜅(𝑦 − 𝑥) , where we have used the monotonicity and subadditivity of 𝜅. This shows that 𝑓 is 𝜅decreasing with 𝐶𝜅 (𝑓) = 1. By Proposition 2.64, 𝑓 ∈ 𝜅𝐵𝑉([0, 1]) with Var𝜅 (𝑓; [0, 1]) ≤ 2. ♥ Our final example, which is left to the reader (Exercise 2.32), shows that the Korenblum variation function V𝑓,𝜅 defined in analogy to (1.13) by V𝑓,𝜅 (𝑥) := Var𝜅 (𝑓; [0, 𝑥])

(0 ≤ 𝑥 ≤ 1)

(2.115)

need not be monotonically increasing. This explains some of the difficulties one en­ counters when replacing Jordan variation by Korenblum variation. There is also a Helly-type selection theorem for 𝜅-decreasing functions; compare this with Theorem 1.11. Theorem 2.67. Let (𝑓𝑛 )𝑛 be a uniformly bounded and uniformly 𝜅-decreasing sequence of functions 𝑓𝑛 : [0, 1] → ℝ. Then there exists a subsequence of (𝑓𝑛 )𝑛 which converges pointwise on [0, 1] to a 𝜅-decreasing function. Proof. The hypotheses mean that 𝐶 := sup max {‖𝑓𝑛 ‖∞ , 𝐶𝜅 (𝑓𝑛 )} < ∞ ; 𝑛

(2.116)

in particular, 𝑓𝑛 (𝑦) − 𝑓𝑛 (𝑥) ≤ 𝐶𝜅(𝑦 − 𝑥)

(0 ≤ 𝑥 < 𝑦 ≤ 1)

(2.117)

for all 𝑛 ∈ ℕ. Since the sequence (𝑓𝑛 )𝑛 is bounded, we can assume, without loss of generality, that the sequence (𝑓𝑛 (𝑥))𝑛 converges²⁰ at each rational point 𝑥 ∈ [0, 1] to some number which we call 𝑓(𝑥). Then the function 𝑓 also satisfies 𝑓(𝑦) − 𝑓(𝑥) ≤ 𝐶𝜅(𝑦 − 𝑥)

(0 ≤ 𝑥 < 𝑦 ≤ 1) ,

(2.118)

at least for rational 𝑥 and 𝑦. We extend 𝑓 to the whole interval [0, 1] by putting 𝑓(𝑥) := lim 𝑓(𝑦𝑛 ) , 𝑛→∞

(2.119)

where (𝑦𝑛 )𝑛 is a rational sequence converging to 𝑥. The fact that this limit exists can be seen as follows. Suppose that 𝑙 := lim inf 𝑓(𝑦) < lim sup 𝑓(𝑦) =: 𝐿 , 𝑦→𝑥−

𝑦→𝑥−

20 To see this, we may use the standard Cantor diagonalization technique.

176 | 2 Nonclassical BV-spaces where 𝑦 ∈ [0, 1] ∩ ℚ, and let (𝑦𝑛 )𝑛 and (𝑦𝑛󸀠 )𝑛 be two rational sequences which converge to 𝑥 and are interlacing in the sense that 𝑦1 < 𝑦1󸀠 < 𝑦2 < 𝑦2󸀠 < . . . < 𝑦𝑛 < 𝑦𝑛󸀠 < . . . < 𝑥 and satisfy lim 𝑓(𝑦𝑛 ) = 𝑙,

𝑛→∞

lim 𝑓(𝑦𝑛󸀠 ) = 𝐿 .

𝑛→∞

Then (2.118) implies that, for 𝑦 = 𝑦𝑛󸀠 and 𝑥 = 𝑦𝑛 , 𝑓(𝑦𝑛󸀠 ) − 𝑓(𝑦𝑛 ) ≤ 𝐶𝜅(𝑦𝑛󸀠 − 𝑦𝑛 )

(𝑛 = 1, 2, 3, . . .) .

Passing to the limit 𝑛 → ∞ yields 𝐿 − 𝑙 ≤ 0, that is, a contradiction. The same argument shows that (2.118) also holds for irrational 𝑥, 𝑦 ∈ [0, 1], which means that 𝑓 is 𝜅-decreasing with 𝐶𝜅 (𝑓) ≤ 𝐶. By Proposition 2.64, 𝑓 has bounded 𝜅variation with Var𝜅 (𝑓; [0, 1]) ≤ 2𝐶, and therefore its discontinuity set 𝐷(𝑓), see (0.49), is at most countable, by (2.108). Using, if necessary, another Cantor diagonalization process, we may assume that the sequence (𝑓𝑛 )𝑛 converges to 𝑓 at each point of 𝐷(𝑓). So, it remains to show that 𝑓𝑛 (𝑥) → 𝑓(𝑥) also at each point of continuity of 𝑓. Let 𝑥0 ∈ [0, 1] \ 𝐷(𝑓) be such a point, and let 𝜀 > 0 be given. Fix 𝑥1 , 𝑥2 ∈ [0, 1] ∩ ℚ such that 𝑥1 < 𝑥0 < 𝑥2 and |𝑓(𝑥𝑖 ) − 𝑓(𝑥0 )|
1, the proof is similar with changes in the constants. Details may be found in [274]. The inclusion (2.145) is proved in the same way as in the special case 𝑘 = 1, see Proposi­ tion 1.38. The spaces 𝑊𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]) have been particularly well-studied for 𝑝 = 1. For example, De la Vallée Poussin [103] has proved the following representation theorem for func­ tions in 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]), which is a natural analogue to Jordan’s representation theorem (Theorem 1.5): Theorem 2.74 (De la Vallée Poussin). A function 𝑓 : [𝑎, 𝑏] → ℝ has bounded second variation Var𝑊 2,1 (𝑓; [𝑎, 𝑏]) if and only if it may be represented in the form 𝑓 = 𝑃𝑓 − 𝑁𝑓 , where both 𝑃𝑓 and 𝑁𝑓 are convex functions. Proof. To prove the “if” part, it obviously suffices to show that every convex function 𝑓 : [𝑎, 𝑏] → ℝ belongs to 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]). Now, if 𝑓 is convex, then the map 𝑓[⋅, 𝑦] : [𝑎, 𝑦) → ℝ defined by 𝑥 󳨃→ 𝑓[𝑥, 𝑦] =

𝑓(𝑦) − 𝑓(𝑥) 𝑦−𝑥

(𝑎 ≤ 𝑥 < 𝑦)

is increasing on [𝑎, 𝑦) for any 𝑦 ∈ (𝑎, 𝑏]. Moreover, the unilateral derivatives 𝑓+󸀠 and 𝑓−󸀠 of 𝑓 exist on [𝑎, 𝑏] and satisfy 𝑓+󸀠 (𝑥) ≤ 𝑓[𝑥, 𝑦] ≤ 𝑓−󸀠 (𝑦)

(𝑎 ≤ 𝑥 < 𝑦)

(2.147)

by definition of the second divided difference in (2.139). Given any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), we have 𝑚−1

Var𝑊 2,1 (𝑓, 𝑃; [𝑎, 𝑏]) = ∑ |𝑓[𝑡𝑗 , 𝑡𝑗+1 ] − 𝑓[𝑡𝑗−1 , 𝑡𝑗 ]| 𝑗=1

𝑚−1

= ∑ 𝑓[𝑡𝑗 , 𝑡𝑗+1 ] − 𝑓[𝑡𝑗−1 , 𝑡𝑗 ] = 𝑓[𝑡𝑚−1 , 𝑏] − 𝑓[𝑎, 𝑡1 ] ≤ 𝑓−󸀠 (𝑏) − 𝑓+󸀠 (𝑎) , 𝑗=1

where we may drop the absolute value after the sum, since 𝑓[⋅, 𝑦] is increasing. Passing to the supremum with respect to all partitions 𝑃 ∈ P([𝑎, 𝑏]) we conclude that 󸀠 󸀠 Var𝑊 2,1 (𝑓; [𝑎, 𝑏]) ≤ 𝑓− (𝑏) − 𝑓+ (𝑎) ,

and so 𝑓 ∈ 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]) which proves the “if” part of the assertion. Now suppose that 𝑓 ∈ 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]), which means that the second variation Var𝑊 2,1 (𝑓; [𝑎, 𝑏]) {𝑚−1 } = sup { ∑ |𝑓[𝑡𝑗 , 𝑡𝑗+1 ] − 𝑓[𝑡𝑗−1 , 𝑡𝑗 ]| : {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏])} { 𝑗=1 }

(2.148)

186 | 2 Nonclassical BV-spaces of 𝑓 on [𝑎, 𝑏] is finite. Without loss of generality we may assume that 𝑓(𝑎) = 0. We use the fact that every function 𝑓 ∈ 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]) is Lipschitz continuous (Exercise 2.44), and so we can find a nullset 𝑁 ⊂ [𝑎, 𝑏] such that 𝑓󸀠 exists on [𝑎, 𝑏] \ 𝑁. We define 𝑝𝑓 , 𝑛𝑓 : [𝑎, 𝑏] \ 𝑁 → ℝ by 𝑝𝑓 (𝑥) :=

1 󸀠 [Var𝑊 2,1 (𝑓; [𝑎, 𝑥]) + 𝑓 (𝑥)] , 2

𝑛𝑓 (𝑥) :=

1 󸀠 [Var𝑊 2,1 (𝑓; [𝑎, 𝑥]) − 𝑓 (𝑥)] , 2

and so we have 𝑝𝑓 − 𝑛𝑓 = 𝑓󸀠 on [𝑎, 𝑏] \ 𝑁. We claim that both functions 𝑝𝑓 and 𝑛𝑓 are increasing. In fact, for 𝑎 ≤ 𝑥 < 𝑦 ≤ 𝑏, we have 𝑊 𝑊 Var𝑊 2,1 (𝑓; [𝑥, 𝑦]) ≤ Var2,1 (𝑓; [𝑎, 𝑦]) − Var2,1 (𝑓; [𝑎, 𝑥]) ,

and hence 𝑊 󸀠 󸀠 2𝑝𝑓 (𝑦) − 2𝑝𝑓 (𝑥) = Var𝑊 2,1 (𝑓; [𝑎, 𝑦]) − Var2,1 (𝑓; [𝑎, 𝑥]) + 𝑓 (𝑦) − 𝑓 (𝑥) 󸀠 󸀠 ≥ Var𝑊 2,1 (𝑓; [𝑥, 𝑦]) + 𝑓 (𝑦) − 𝑓 (𝑥) ≥ 0 ,

and similarly for the function 𝑛𝑓 . Since both functions 𝑝𝑓 and 𝑛𝑓 are increasing and bounded on [𝑎, 𝑏] \ 𝑁, we may extend them to increasing bounded functions on the whole interval [𝑎, 𝑏] which we still denote by 𝑝𝑓 and 𝑛𝑓 . It is now a natural idea to define 𝑃𝑓 , 𝑁𝑓 : [𝑎, 𝑏] → ℝ by 𝑥

𝑥

𝑃𝑓 (𝑥) := ∫ 𝑝𝑓 (𝑡) 𝑑𝑡,

𝑁𝑓 (𝑥) := ∫ 𝑛𝑓 (𝑡) 𝑑𝑡 .

𝑎

𝑎

Then we obtain 𝑥

𝑥

𝑃𝑓 (𝑥) − 𝑁𝑓 (𝑥) = ∫ [𝑝𝑓 (𝑡) − 𝑛𝑓 (𝑡)] 𝑑𝑡 = ∫ 𝑓󸀠 (𝑡) 𝑑𝑡 = 𝑓(𝑥) 𝑎

𝑎

since 𝑓(𝑎) = 0. Moreover, both functions 𝑃𝑓 and 𝑁𝑓 are convex because their deriva­ tives are monotonically increasing, and so we have proved the “only if” part of the assertion. Note that we have constructed the convex functions 𝑃𝑓 and 𝑁𝑓 satisfying 𝑃𝑓 − 𝑁𝑓 = 𝑓 in Theorem 2.74 from the increasing functions 𝑝𝑓 and 𝑛𝑓 satisfying 𝑝𝑓 − 𝑛𝑓 = 𝑓󸀠 for the derivative (up to nullsets). Thus, one might ask if there is a connection between func­ tions belonging to 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]) and functions having derivatives in 𝑊𝐵𝑉1,1 ([𝑎, 𝑏]) = 𝐵𝑉([𝑎, 𝑏]). In fact, we have seen that every function 𝑓 ∈ 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]) admits unilat­ eral derivatives 𝑓+󸀠 and 𝑓−󸀠 on [𝑎, 𝑏]. Moreover, in [274], it is shown that there exists a nullset 𝑁 ⊂ [𝑎, 𝑏] such that 𝑓 ∈ 𝐶1 ([𝑎, 𝑏] \ 𝑁). This means that the derivative 𝑓󸀠 of a function 𝑓 ∈ 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]) exists and is continuous a.e. on [𝑎, 𝑏]. One cannot claim, however, that the derivative 𝑓󸀠 is even absolutely continuous a.e. on [𝑎, 𝑏], as the fol­ lowing example shows.

2.7 Comments on Chapter 2

| 187

Example 2.75. Let 𝜑 : [0, 1] → ℝ be the classical Cantor function, see Example 3.6 in the next chapter, and let 𝑓 : [0, 1] → ℝ be defined by 𝑥

𝑓(𝑥) := ∫ 𝜑(𝑡) 𝑑𝑡

(0 ≤ 𝑥 ≤ 1) .

(2.149)

0

Since 𝜑 is monotonically increasing, 𝑓 is convex, and so 𝑓 ∈ 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]), by Theorem 2.74. On the other hand, we have 𝑓󸀠 = 𝜑 a.e. on [0, 1], and 𝜑 is not absolutely continuous because it fails to have the so-called Luzin property, as we will also show in the next chapter. ♥ Remarkably, if we require a function 𝑓 ∈ 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]) to have an absolutely contin­ uous derivative, we get an interesting integral representation for the second variation of 𝑓 which was proved by Russell in [276]. Theorem 2.76 (Russell). Suppose that a function 𝑓 ∈ 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]) has an absolutely continuous derivative 𝑓󸀠 a.e. on [𝑎, 𝑏]. Then the second variation of 𝑓 has the integral representation 𝑏

Var𝑊 2,1 (𝑓; [𝑎, 𝑏])

= ∫ |𝑓󸀠󸀠 (𝑡)| 𝑑𝑡 .

(2.150)

𝑎

We postpone the proof of Theorem 2.76 to Chapter 3, where we will give a more general result (Theorem 3.39). Theorem 2.76 shows again that the function (2.149) from Example 2.75 cannot have an absolutely continuous derivative. Indeed, for 𝑓 as in (2.149), the left-hand side of (2.150) is 1, while the right-hand side of (2.150) is 0 because 𝑓󸀠 (𝑡) = 𝜑(𝑡) = 0 a.e. on [0, 1].

2.7 Comments on Chapter 2 Among the various generalized variations discussed in this chapter, the Waterman variation (2.26) is probably the most important one, both from its theoretical interest and from an application-oriented point of view. As pointed out several times before, passing from the Wiener space 𝑊𝐵𝑉𝑝 to the Wiener–Young space 𝑊𝐵𝑉𝜙 has the same advantages and drawbacks as passing from the Lebesgue space 𝐿 𝑝 to the Orlicz space 𝐿 𝜙 : the range of applications increases, but the technical expenditure increases as well. It is not surprising that all functions 𝑓 ∈ 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]), although not being contin­ uous, are always regular in the sense of our definition in Section 0.3; so they share this important property with classical 𝐵𝑉-functions (in particular, monotone functions). The following Theorem [255] gives a refinement (and certain converse) of this: every function 𝑓 ∈ 𝑅([𝑎, 𝑏]) is contained in some space 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) for a suitable Young function 𝜙.

188 | 2 Nonclassical BV-spaces Theorem 2.77. The equalities ⋃ 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) = 𝑅([𝑎, 𝑏]), 𝜙

⋂ 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) = 𝐹𝑅([𝑎, 𝑏])

(2.151)

𝜙

hold where 𝑅([𝑎, 𝑏]) denotes the space of all regular functions, 𝐹𝑅([𝑎, 𝑏]) denotes the space of all regular functions which assume only a finite number of values, and the in­ tersection and union in (2.151) are taken over all Young functions 𝜙. Many interesting structural theorems about 𝑊𝐵𝑉𝜙 -functions have been given by Prus–Wiśniowski in a series of papers [254–261]. For example, in [256], the author analyzes the class of all functions 𝑓 ∈ 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) satisfying lim sup {Var𝑊 𝜙 (𝑓, 𝑃; [𝑎, 𝑏]) : 𝜇(𝑃) ≤ 𝛿} = 0 ,

(2.152)

𝛿→0+

where 𝜇(𝑃) denotes the mesh size (1.2) of the partition 𝑃. This is analogous to what we have done in the first part of Section 1.2 for classical 𝐵𝑉-functions. In [99], a nec­ essary and sufficient condition is given under which a sequence of functions (ℎ𝑛 ∘ 𝑓)𝑛 is bounded, where 𝑓 ∈ 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) with 𝑓([𝑎, 𝑏]) ⊆ [𝑐, 𝑑], and ℎ𝑛 ∈ 𝑊𝐵𝑉𝜓 ([𝑐, 𝑑]) for each 𝑛 ∈ ℕ. The spaces 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) have been introduced and studied by Riesz [265], and the more general space 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) by Medvedev [211]. Both spaces will play a prominent role in Section 3.5 in the next chapter. One of the important properties of the Riesz space 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) is its close relation to (scalar) Sobolev spaces; we shall discuss this in the next chapter. In Proposition 2.51, we have shown that 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) is a Banach algebra with respect to multiplication. The following elementary counterexample shows that 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) is not stable with respect to the composition of functions. Example 2.78. For 0 < 𝜏 < 1, define 𝑓𝜏 : [0, 1] → ℝ by 𝑓𝜏 (𝑥) := 𝑥𝜏 . A somewhat cumbersome calculation²⁴ shows that 𝑓𝜏 ∈ 𝑅𝐵𝑉𝑝 ([0, 1]) ⇐⇒ moreover, ‖𝑓𝜏 ‖𝑅𝐵𝑉𝑝 = in this case. So, if we take 1 − 𝜏 < 𝑅𝐵𝑉𝑝 ([0, 1]).

1 𝑝

1 > 1 −𝜏; 𝑝

(2.153)

𝜏 [1 − (1 − 𝜏)𝑝]1/𝑝

≤ 1 − 𝜏2 , then 𝑓𝜏 ∈ 𝑅𝐵𝑉𝑝 ([0, 1]), but 𝑓𝜏 ∘ 𝑓𝜏 = 𝑓𝜏2 ∈ ̸ ♥

Another example of this kind will be given in Chapter 5 (Example 5.8). The following proposition due to Szigeti [299, 300] shows how to choose the “right” indices 𝑝, 𝑞,

24 In the next chapter, we will provide a much more elegant method to prove (2.153) and to calculate ‖𝑓𝜏 ‖𝑅𝐵𝑉𝑝 , see Example 3.35.

2.7 Comments on Chapter 2 |

189

and 𝑟 to guarantee that compositions of Riesz space functions remain in some other Riesz space. An analogous result for several variables has been proved in [216]. Proposition 2.79. Suppose that 𝑝, 𝑞, 𝑟 ∈ (1, ∞) satisfy (1 −

1 1 1 ) (1 − ) = 1 − . 𝑝 𝑞 𝑟

(2.154)

Let 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) be monotone with 𝑓([𝑎, 𝑏]) ⊆ [𝑐, 𝑑], and 𝑔 ∈ 𝑅𝐵𝑉𝑞 ([𝑐, 𝑑]). Then 𝑔 ∘ 𝑓 ∈ 𝑅𝐵𝑉𝑟 ([𝑎, 𝑏]) and Var𝑅𝑟 (𝑔 ∘ 𝑓; [𝑎, 𝑏]) ≤ 𝐶 Var𝑅𝑞 (𝑔; [𝑎, 𝑏])𝑟/𝑞 Var𝑅𝑝 (𝑓; [𝑎, 𝑏])1−𝑟/𝑞 for some 𝐶 > 0. Example 2.78 shows that condition (2.154) is sharp. In fact, by choosing 𝑓 := 𝑓𝛼 ∈ 𝑅𝐵𝑉𝑝 ([0, 1]) and 𝑔 := 𝑓𝛽 ∈ 𝑅𝐵𝑉𝑞 ([0, 1]) (0 < 𝛼, 𝛽 < 1), we obtain, by (2.153), (1 −

1 1 ) (1 − ) < 𝛼𝛽 𝑝 𝑞

as a necessary and sufficient condition, and 𝑔 ∘ 𝑓 = 𝑓𝛼𝛽 implies that 1−

1 < 𝛼𝛽 𝑟

to ensure that 𝑔 ∘ 𝑓 ∈ 𝑅𝐵𝑉𝑟 ([0, 1]). Passing from the Riesz space 𝑅𝐵𝑉𝑝 to the Riesz–Medvedev space 𝑅𝐵𝑉𝜙 , we of course encounter similar phenomena. We will come back to both classes of spaces in Section 3.5. Many interesting facts about the space 𝑅𝐵𝑉𝜙 are discussed in [209]. Proposition 2.57 shows that when studying the space 𝑅𝐵𝑉𝜙 , we have to assume that 𝜙 ∈ ∞1 if we want to obtain something new. In this connection, the following question is of interest: Problem 2.1. Suppose that 𝜙 is a Young function satisfying 𝜙 ∈ ̸ ∞𝑝 for some 𝑝 > 1. Does this imply that 𝑅𝐵𝑉𝜙 = 𝑅𝐵𝑉𝑝 ? In view of their importance, let us now spend some time to comment on our results about the Waterman spaces Λ𝐵𝑉 and Λ 𝑞 𝐵𝑉. The proof of Proposition 2.17 (f) shows that convergence in the norm (2.30) implies uniform convergence on [𝑎, 𝑏]. Thus, the set 𝐶([𝑎, 𝑏]) ∩ Λ𝐵𝑉([𝑎, 𝑏]) of continuous functions in Λ𝐵𝑉([𝑎, 𝑏]) forms a closed sub­ space of Λ𝐵𝑉([𝑎, 𝑏]), and thus is also a Banach space with respect to the norm (2.30). The space Λ𝐵𝑉 has been introduced by Waterman in [315, 316] and afterwards has been studied and further developed by several authors. Without aiming for a complete list, we mention [24, 25, 213, 214, 246, 247, 250, 251, 260, 262, 287, 288, 312, 313].

190 | 2 Nonclassical BV-spaces We remark that the definition of Λ𝐵𝑉 is somewhat different in the literature: the sequence Λ = (𝜆 𝑛 )𝑛 is supposed to be increasing and unbounded, (2.24) is replaced by ∞

1 = ∞, 𝑛=1 𝜆 𝑛 ∑

and all expressions of the form |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| are not multiplied, but divided by 𝜆 𝑘 . The equalities (2.33) have been proved by Perlman in [246], where it is also shown that (2.33) is not true if the intersection and union are taken only over a countable family of Waterman sequences Λ. The second equality in (2.33) shows that a function 𝑓 ∈ Λ𝐵𝑉 only has removable discontinuities or jumps. Perlman and Waterman [247] prove that whenever lim inf 𝑓(𝑥) ≤ 𝑓(𝑥0 ) ≤ lim sup 𝑓(𝑥) 𝑥→𝑥0

𝑥→𝑥0

at each point of discontinuity 𝑥0 of 𝑓, then the total Waterman variation (2.26) of 𝑓 is actually independent of the value 𝑓(𝑥0 ). Our Proposition 2.28 which gives a neces­ sary and sufficient condition for inclusion between different Waterman spaces is also proved in the paper [247]. In this connection, the following result [262] is interesting: for every Waterman sequence Λ = (𝜆 𝑛 )𝑛 , one may find a Waterman sequence 𝑀 = (𝜇𝑛 )𝑛 (called “regularization” of Λ) such that lim sup 𝑛→∞

𝜇𝑛+1 =1 𝜇𝑛

and (𝜆[1, 𝑛])𝑛 ∼ (𝜇[1, 𝑛])𝑛 (in the notation (2.42)), and hence Λ𝐵𝑉 = 𝑀𝐵𝑉. The Proposi­ tions 2.32 and 2.33, as well as the idea of using zigzag functions to separate the function classes 𝑊𝐵𝑉𝑝 and Λ 𝑞 𝐵𝑉, are due to Pierce and Velleman [250]. The subspace Λ𝑐 𝐵𝑉 of Λ𝐵𝑉 has been introduced in 1976 by Waterman [315, 316]. For 𝜆 𝑛 ≡ 1 (i.e. Λ𝐵𝑉 = 𝐵𝑉), the space Λ𝑐 𝐵𝑉 is not interesting because it only contains constant functions. However, if Λ = (𝜆 𝑛 )𝑛 is not bounded away from zero, the following question seems to be important: Problem 2.2. How can we characterize the elements of Λ𝑐 𝐵𝑉 in case 𝜆 𝑛 → 0 (i.e. Λ𝐵𝑉 ≠ 𝐵𝑉)? We mentioned that an analogous subspace of the Schramm space 𝛷𝐵𝑉 has been de­ fined in the literature, namely, the class 𝛷𝑐 𝐵𝑉([𝑎, 𝑏]) of all 𝑓 ∈ 𝛷𝐵𝑉([𝑎, 𝑏]) satisfying lim Var𝛷 (𝑓/𝜆; [𝑎, 𝑏])) = 0 ,

𝜆→∞

where Var𝛷 (𝑓; [𝑎, 𝑏]) denotes the Schramm variation (2.71). Here, a similar question arises: Problem 2.3. How can we characterize the elements of 𝛷𝑐 𝐵𝑉 for arbitrary Schramm sequences 𝛷?

2.7 Comments on Chapter 2

| 191

Observe that, as Exercise 2.1 shows, this question does not make sense in the Wiener– Young space 𝑊𝐵𝑉𝜙 (which may be considered as a Schramm space with 𝜙𝑛 (𝑡) ≡ 𝜙(𝑡)) because every function 𝑓 ∈ 𝑊𝐵𝑉𝜙 is continuous in the Wiener–Young variation (2.2). Interestingly, the Waterman space Λ𝐵𝑉 is also intimately related to the general­ ized Hölder space 𝐿𝑖𝑝𝜔,𝑝 introduced in Definition 0.54. Imbedding theorems between these spaces (in both directions) have been established by various authors; for exam­ ple, the following result ([133], see also [312]) holds. Proposition 2.80. Let Λ = (𝜆 𝑛 )𝑛 be a Waterman sequence, 𝜔 : [0, ∞) → [0, 𝜔) a mod­ ulus of continuity, and 1 ≤ 𝑝 < ∞. Then the inclusion Λ𝐵𝑉 ⊆ 𝐿𝑖𝑝𝜔,𝑝 holds if and only if 𝑘1/𝑝 1 max = 𝑂 (𝑛1/𝑝 𝜔 ( )) (𝑛 → ∞) , 𝑘=1,...,𝑛 𝜆[1, 𝑘] 𝑛 where we have used the notation (2.42). In particular, for 0 < 𝑞 < 1 and 0 < 𝛼 ≤ 1, we have Λ 𝑞 𝐵𝑉 ⊆ 𝐿𝑖𝑝𝛼,𝑝 if and only if 1 𝛼 ≤ min { , 1 − 𝑞} , 𝑝 where Λ 𝑞 𝐵𝑉 is given in Definition 2.29, and 𝐿𝑖𝑝𝛼,𝑝 in Definition 0.54. Imbedding theorems for Hölder spaces into spaces of functions of (generalized) bounded variation have also been given in the literature. The simplest example is Proposition 1.34 which states that 𝐿𝑖𝑝1/𝑝 ([𝑎, 𝑏]) 󳨅→ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) for 1 ≤ 𝑝 < ∞. The following more general and precise result involving the space 𝐿𝑖𝑝𝜔,∞ ([𝑎, 𝑏]) from Definition 0.54 has been proved by Medvedev in [212], see also [213]. Proposition 2.81. Let 𝜙 be a Young function which satisfies the 𝛿2 -condition (2.4). Then 𝐿𝑖𝑝𝜔,∞ ([𝑎, 𝑏]) 󳨅→ 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) if and only if 𝜔(𝑡) = 𝑂(𝜙−1 (𝑡))

(𝑡 → 0+) .

(2.155)

In the special case 𝜔(𝑡) = 𝑡𝛼 (and hence 𝐿𝑖𝑝𝜔,∞ = 𝐿𝑖𝑝𝛼 ) and 𝜙(𝑡) = 𝑡𝑝 (and hence 𝑊𝐵𝑉𝜙 = 𝑊𝐵𝑉𝑝 ), condition (2.155) holds precisely for 𝛼 ≥ 1/𝑝. Therefore, we re­ cover Proposition 1.34 for the sufficiency of (2.155) for the imbedding 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) 󳨅→ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]). Observe also that Exercises 1.59 and 1.60 are contained in Proposi­ tion 2.79 for the special choice 𝜙(𝑡) = 𝑡𝑝 . The following result which provides a certain converse of Proposition 2.81 is also due to Medvedev [212]: Proposition 2.82. Let 𝜔 : [0, ∞) → [0, ∞) be a modulus of continuity, and let Λ = (𝜆 𝑛 )𝑛 be a Waterman sequence. Then the inclusion 𝐿𝑖𝑝𝜔,∞ ⊆ Λ𝐵𝑉 holds if and only if there exists a nonnegative sequence (𝑡𝑛 )𝑛 satisfying ∞

∞

∑ 𝑡𝑛 ≤ 1,

∑ 𝜆 𝑛𝜔(𝑡𝑛 ) < ∞ .

𝑛=1

𝑛=1

(2.156)

192 | 2 Nonclassical BV-spaces Again, let us check what Proposition 2.82 means for 𝜔(𝑡) = 𝑡𝛼 (and hence 𝐿𝑖𝑝𝜔,∞ = 𝐿𝑖𝑝𝛼 ) and 𝜆 𝑛 = 𝑛−𝑞 (and hence Λ𝐵𝑉 = Λ 𝑞 𝐵𝑉). Choosing in this case 𝑡𝑛 := 2−𝑛 , the second condition in (2.156) reads ∞

1 < ∞, 𝑞 2𝑛𝛼 𝑛 𝑛=1

(2.157)

∑

and the classical quotient criterion implies that (2.157) holds for any 𝑞 ≥ 0 and 𝛼 > 0. So, we have the imbedding 𝐿𝑖𝑝𝛼 󳨅→ Λ 𝑞 in this case.²⁵ More general sufficient condi­ tions for imbeddings of type 𝐿𝑖𝑝𝜔,∞ 󳨅→ Λ𝐵𝑉 (which are sometimes also necessary) may be found in [213]. As we have shown in Proposition 1.34, the Wiener space 𝑊𝐵𝑉1/(1−𝑞) contains the Hölder space 𝐿𝑖𝑝1−𝑞 , see (1.68). However, the following example shows that the smaller space Λ 𝑞 𝐵𝑉([0, 1]) does not contain this Hölder space. Example 2.83. This example is quite similar to Example 1.24, where we constructed a function which belongs to 𝐿𝑖𝑝𝛼 for any 𝛼 < 1, but not to 𝐵𝑉. Given 𝑞 ∈ (0, 1), let ∞

1 , 𝑘 log (𝑘 + 1) 𝑘=1

𝛾 := 𝜁(1, 𝑞) = ∑

𝑞

𝑡𝑛 :=

1 1 𝑛 , ∑ 𝛾 𝑘=1 𝑘 log𝑞 (𝑘 + 1)

and define 𝑓 : [0, 1] → ℝ by 0 { { { { { { 1𝑞 ∑𝑛𝑘=1 𝑓(𝑥) := { 𝛾1 ∞ { 𝑞∑ { 𝑘=1 { 𝛾 { { linear {

for 𝑥 = 0 , (−1)𝑘+1 𝑘𝑞 log(𝑘+1) (−1)𝑘+1 𝑘𝑞 log(𝑘+1)

for 𝑥 = 𝑡𝑛 , for 𝑥 = 1 , otherwise .

As in Example 1.24, one may show that 𝑓 ∈ 𝐿𝑖𝑝1−𝑞 ([0, 1]); we claim that 𝑓 ∈ ̸ Λ 𝑞𝐵𝑉([0, 1]). In fact, on the interval [𝑎𝑘 , 𝑏𝑘 ] := [𝑡𝑘−1 , 𝑡𝑘 ], we have 𝑛

𝑘−1 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| (−1)𝑗+1 (−1)𝑗+1 ] 1 𝑛 [𝑘 − = ∑ ∑ ∑ 𝑞 𝑞 𝑝 𝑞 𝑘 𝛾 𝑘=1 𝑗=1 𝑗 log(𝑗 + 1) 𝑗=1 𝑗 log(𝑗 + 1) 𝑘=1 ] [ 𝜁(1, 𝑞) 1 1 𝑛 = → ∞ (𝑛 → ∞) , = 𝑞 ∑ 𝑞 𝛾 𝑘=1 𝑘 log(𝑘 + 1) 𝛾𝑞

∑

and thus VarΛ 𝑞 (𝑓; [0, 1]) = ∞ and so 𝑓 ∈ ̸ Λ 𝑞 𝐵𝑉([0, 1]).

♥

We remark that sufficient conditions for the inclusions 𝐶([𝑎, 𝑏]) ⊆ Λ𝑐 𝐵𝑉([𝑎, 𝑏]),

𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) ⊆ Λ𝑐𝑞 𝐵𝑉([𝑎, 𝑏])

25 Observe that we may combine (1.68) and (2.46) to deduce that 𝐿𝑖𝑝𝛼 󳨅→ 𝑊𝐵𝑉1/𝛼 󳨅→ Λ 𝑞 ; however, according to Proposition 2.32, the last imbedding is true only for 𝑞 > 𝑝1󸀠 = 1 − 𝛼, while (2.157) does not require a link between 𝛼 and 𝑞.

2.7 Comments on Chapter 2

| 193

are given in Exercises 2.12 and 2.13. Relations between spaces of functions of bounded variation, generalized Hölder spaces, and Orlicz spaces are studied in the survey paper [101]. In Proposition 1.27, we have shown how bounded variation of 𝑓 reflects in terms of the corresponding Banach indicatrix 𝐼𝑓 : we have 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) if and only if 𝐼𝑓 ∈ 𝐿 1 (ℝ). This admits an interesting generalization which provides a link between mod­ uli of continuity, Waterman sequences, and the Banach indicatrix. Suppose that 𝜔 : [0, ∞) → [0, ∞) is a modulus of continuity which satisfies 𝜔(𝑡) = 𝑜(𝑡) as 𝑡 → ∞. Then the sequence Λ = (𝜆 𝑛 )𝑛 defined by 𝜆 𝑛 := 𝜔(𝑛) − 𝜔(𝑛 − 1) is a Waterman sequence.²⁶ In [25, Theorem 5.2], it is shown that the condition ∞

∫ 𝜔(𝑁𝑓 (𝑦)) 𝑑𝑦 < ∞ −∞

for a regular function 𝑓 : [𝑎, 𝑏] → ℝ implies that 𝑓 ∈ Λ𝑐 𝐵𝑉([𝑎, 𝑏]), where Λ is as before. A particularly interesting case is 𝜔(𝑡) = 𝑡𝛼 with 0 < 𝛼 < 1; here, the condition²⁷ ∞

∫ 𝑁𝑓 (𝑦)𝛼 𝑑𝑦 < ∞ −∞

Λ𝑐1−𝛼 𝐵𝑉([𝑎, 𝑏]).

By Proposition 2.33, this, in turn, implies 𝑓 ∈ implies that 𝑓 ∈ 𝑊𝐵𝑉1/𝛼 ([𝑎, 𝑏]) which is a result by Zerekidze [328]. In this chapter, we have discussed various imbedding theorems between the Wiener–Young space 𝑊𝐵𝑉𝜙 , the generalized Hölder space 𝐿𝑖𝑝𝜔,∞ , the Waterman space Λ𝐵𝑉, and the Chanturiya class V𝜈 . It is always illuminating to illustrate such imbeddings by means of the special choices 𝜙(𝑡) = 𝑡𝑝 ,

𝜔(𝑡) = 𝑡𝛼 ,

𝜆𝑛 =

1 , 𝑛𝑞

𝜈𝑛 = 𝑛𝑟

which lead to the function classes 𝑊𝐵𝑉𝑝 , 𝐿𝑖𝑝𝛼 , Λ 𝑞 𝐵𝑉, and V𝜈𝑟 , respectively. In Ta­ ble 2.7 below, we summarize our previous results for these special cases. In this table, we tacitly assume that 1 ≤ 𝑝 < ∞, 0 < 𝛼 ≤ 1, 0 < 𝑞 ≤ 1, and 0 < 𝑟 ≤ 1. Note that the class 𝑊𝐵𝑉𝑝 is increasing in 𝑝, the class 𝐿𝑖𝑝𝛼 is decreasing in 𝛼, the class Λ 𝑞 𝐵𝑉 is increasing in 𝑞, and the class V𝜈𝑟 is increasing in 𝑟. While the Waterman spaces Λ𝐵𝑉 are extremely useful in the theory of Fourier series (see Section 7.2 below), the Schramm space 𝛷𝐵𝑉 seems to be only of very limited interest. However, Proposition 2.43 shows that it provides a unified approach to many spaces we considered so far. A particularly simple example of a Schramm space is

26 Indeed, from the concavity of 𝜔, it follows that (𝜆 𝑛 )𝑛 is decreasing, from 𝜔(𝑛) = 𝑜(𝑛) as 𝑛 → ∞ it follows that (𝜆 𝑛 )𝑛 tends to zero, and from 𝜆[1, 𝑛] = 𝜔(𝑛) → ∞ as 𝑛 → ∞ it follows that (2.24) is true. 27 Observe that our hypothesis 𝜔(𝑡) = 𝑜(𝑡) as 𝑡 → ∞ excludes the case 𝛼 = 1.

194 | 2 Nonclassical BV-spaces Table 2.7. Imbeddings between function classes. The function class

is imbedded into

for

𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) Λ 𝑞 ([𝑎, 𝑏]) 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) V𝜈𝑟 ([𝑎, 𝑏]) Λ 𝑞 𝐵𝑉([𝑎, 𝑏])

𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) Λ 𝑞 𝐵𝑉([𝑎, 𝑏]) V𝜈𝑟 ([𝑎, 𝑏]) Λ𝑐𝑞 𝐵𝑉([𝑎, 𝑏]) V𝜈𝑟 ([𝑎, 𝑏])

𝑝𝛼 ≤ 1 𝑞 ≤ 1 − 1/𝑝 𝑞 > 1 − 1/𝑝 𝑟 ≥ 1 − 1/𝑝 𝑞>𝑟 𝑞≤𝑟

when 𝛷 is a constant sequence, i.e. 𝜙𝑛 (𝑡) ≡ 𝜙(𝑡) for some Young function 𝜙. In this case the variation (2.70) has the form ∞

Var𝛷 (𝑓, 𝑆∞ ) = Var𝛷 (𝑓, 𝑆∞ ; [𝑎, 𝑏]) = ∑ 𝜙(|𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|), 𝑘=1

and the corresponding space 𝛷𝐵𝑉([𝑎, 𝑏]) is sometimes denoted by 𝜙𝐵𝑉([𝑎, 𝑏]). Ob­ serve, however, that in the definition of this space, we may restrict ourselves to finite collections 𝑆 ∈ 𝛴([𝑎, 𝑏]). This means that we may define, in analogy to (2.73), the vari­ ation by Var𝛷 (𝑓; [𝑎, 𝑏]) = sup {Var𝛷 (𝑓, 𝑆; [𝑎, 𝑏]) : 𝑆 ∈ 𝛴([𝑎, 𝑏])} 𝑛

= sup { ∑ 𝜙(|𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|) : {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏])} . 𝑘=1

The proof of formula (2.54) can be found in [80, Theorem 1], the proof of formula (2.55) in [79, Theorem 4]. The related Proposition 2.36 has been proved in [24]. Our Propositions 2.38 and 2.39, as well as Examples 2.40 and 2.41, are taken from [25]. Interestingly, the discontinuity behavior of a function 𝑓 : [𝑎, 𝑏] → ℝ can be described by means of the characteristic (2.52). In fact, in [79] (see also [80, 81]), it is shown that in order for 𝑓 to only have removable discontinuities or discontinuities of the first kind (jumps), it is necessary and sufficient that 𝜈(𝑓)𝑛 = 𝑜(𝑛)

(𝑛 → ∞);

compare this with (2.53). Since 1 lim 𝜙−1 ( ) = 0 𝑛

𝑛→∞

and

𝑛

lim ∑ 𝜆 𝑘 = ∞ ,

𝑛→∞

𝑘=1

Table 2.2 shows that this is true for functions from 𝑊𝐵𝑉𝜙 and Λ𝐵𝑉, and thus also for all special cases like 𝐵𝑉, 𝑊𝐵𝑉𝑝 , or Λ 𝑞 𝐵𝑉.

2.7 Comments on Chapter 2

| 195

The following definition [287] gives a unified approach to the Wiener space 𝑊𝐵𝑉𝜙 introduced in Definition 2.2 and the Waterman space Λ𝐵𝑉 introduced in Defini­ tion 2.15. Later, this class of functions was studied again by Kim [158] who seems to have been unaware of the paper [287]. Definition 2.84. Let 𝜙 : [0, ∞) → [0, ∞) be a Young function and Λ = (𝜆 𝑛)𝑛 be a Waterman sequence. Given a function 𝑓 : [𝑎, 𝑏] → ℝ and a collection 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]), the positive real number ∞

Var𝜙,Λ (𝑓, 𝑆∞ ) = Var𝜙,Λ (𝑓, 𝑆∞ ; [𝑎, 𝑏]) := ∑ 𝜆 𝑘 𝜙(|𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|)

(2.158)

𝑘=1

is called the (𝜙, Λ)-variation of 𝑓 on [𝑎, 𝑏] with respect to 𝑆∞ , while the (possibly infi­ nite) number Var𝜙,Λ (𝑓) = Var𝜙,Λ (𝑓; [𝑎, 𝑏]) := sup {Var𝜙,Λ (𝑓, 𝑆∞ ; [𝑎, 𝑏]) : 𝑆∞ ∈ 𝛴∞ ([𝑎, 𝑏])} , where the supremum is taken over all collections 𝑆∞ ∈ 𝛴∞ ([𝑎, 𝑏]), is called the total (𝜙, Λ)-variation of 𝑓 on [𝑎, 𝑏]. In case Var𝜙,Λ (𝑐𝑓; [𝑎, 𝑏]) < ∞, for some 𝑐 > 0, we say that 𝑓 has bounded (𝜙, Λ)-variation (or Schramm–Waterman variation) on [𝑎, 𝑏] and write ◼ 𝑓 ∈ Λ𝐵𝑉𝜙 ([𝑎, 𝑏]). Clearly, Definition 2.84 contains many of the spaces considered before. Thus, choosing 𝜙(𝑡) = 𝑡 in (2.158), we get Var𝜙,Λ (𝑓; [𝑎, 𝑏]) = VarΛ (𝑓; [𝑎, 𝑏]) , and hence Λ𝐵𝑉𝜙 = Λ𝐵𝑉. By choosing 𝜆 𝑘 ≡ 1 in (2.158), we get Var𝜙,Λ (𝑓; [𝑎, 𝑏]) = Var𝑊 𝜙 (𝑓; [𝑎, 𝑏]) , and hence Λ𝐵𝑉𝜙 = 𝑊𝐵𝑉𝜙 . Moreover, combining the proof of Proposition 2.36 (a) and (b), one may prove the following connection of the space Λ𝐵𝑉𝜙 ([𝑎, 𝑏]) with the Chanturiya class V𝜈 ([𝑎, 𝑏]), see [25]. Proposition 2.85. For any Young function 𝜙 and any Waterman sequence Λ, the inclu­ sion Λ𝐵𝑉𝜙 ([𝑎, 𝑏]) ⊆ V𝜈 ([𝑎, 𝑏]) (2.159) holds true for 𝜈𝑛 := 𝑛𝜙−1 (

1 ), 𝜆[1, 𝑛]

where 𝜆[1, 𝑛] = 𝜆 1 + . . . + 𝜆 𝑛 . Observe that Proposition 2.85 contains all inclusions occurring in Table 2.2 as special cases. In fact, for arbitrary 𝜙 and 𝜆 𝑛 ≡ 1, we get 1 𝜈𝑛 = 𝑛𝜙−1 ( ) , 𝑛

196 | 2 Nonclassical BV-spaces in accordance with the third row in Table 2.2, while for arbitrary Λ and 𝜙(𝑡) = 𝑡, we get 𝜈𝑛 =

𝑛 , 𝜆[1, 𝑛]

in accordance with the fourth row in Table 2.2. Spaces of functions of bounded variation have also been considered with weight. A quite general definition is given in [27], where the authors define, for a given strictly increasing weight function 𝑤 : [𝑎, 𝑏] → ℝ+ and some Young function 𝜙 : [0, ∞) → [0, ∞), the weighted 𝜙-variation of 𝑓 : [𝑎, 𝑏] → ℝ in Riesz’s sense by |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| {𝑚 } ) (𝑤(𝑡𝑗 ) − 𝑤(𝑡𝑗−1 ))} , Var𝑅𝜙,𝑤 (𝑓; [𝑎, 𝑏]) := sup { ∑ 𝜙 ( 𝑤(𝑡 ) − 𝑤(𝑡 ) 𝑗 𝑗−1 {𝑗=1 }

(2.160)

where the supremum is taken over all partitions {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]). The corre­ sponding space of functions of bounded weighted 𝜙-variation is denoted by 𝑅𝐵𝑉 𝜙,𝑤 ([𝑎, 𝑏]). Still, more generally, Chistyakov [87] replaces (2.160) by |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| {𝑚 } ) (𝑤(𝑡𝑗 ) − 𝑤(𝑡𝑗−1 ))} , Var𝛷,𝑤 (𝑓; [𝑎, 𝑏]) := sup { ∑ 𝜙𝑗 ( 𝑤(𝑡 ) − 𝑤(𝑡 ) 𝑗 𝑗−1 {𝑗=1 } with 𝛷 = (𝜙𝑛 )𝑛 being a Schramm sequence, see Definition 2.42. Equipped with the norm ‖𝑓‖𝛷𝐵𝑉𝑤 := |𝑓(𝑎)| + inf {𝜆 > 0 : Var𝛷,𝑤 (𝑓/𝜆; [𝑎, 𝑏])) ≤ 1}, the corresponding set 𝛷𝐵𝑉𝑤 ([𝑎, 𝑏]) of functions of bounded weighted Schramm vari­ ation then becomes a Banach space. Given a Schramm sequence 𝛷 = (𝜙𝑛 )𝑛 , some kind of second Schramm variation for 𝑓 ∈ 𝐵([𝑎, 𝑏]) is defined by ∞

Var2,𝛷 (𝑓) = Var2,𝛷 (𝑓; [𝑎, 𝑏]) := sup { ∑ 𝜙𝑘 (|𝑓[𝑡𝑘+1 , 𝑡𝑘+2 ] − 𝑓[𝑡𝑘 − 𝑡𝑘+1 ]|)} , 𝑘=1

with the supremum being taken over all collections 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]), and studied in the recent paper [121]. Of course, this is inspired by the Wiener variation Var𝑊 2,1 (𝑓) introduced in Definition 2.71. The paper [121] also contains an integral representation for functions with finite second Schramm variation; this generalizes formula (2.150) from Theorem 2.76. The Korenblum variation Var𝜅 (𝑓; [𝑎, 𝑏]) and the Banach space 𝜅𝐵𝑉([𝑎, 𝑏]) have been introduced in 1975 by Korenblum [163] in connection with the Poisson integral representation of certain classes of harmonic functions on the complex unit disc. In our presentation in Section 2.5, we mainly followed the paper [102] which contains the main decomposition theorem and a Helly-type selection principle for 𝜅𝐵𝑉-functions. Further results in this direction may be found in [158] and [161]. Given a distortion

2.7 Comments on Chapter 2

| 197

function 𝜅 : [0, 1] → [0, 1] and a Schramm sequence 𝛷 = (𝜙𝑛 )𝑛 , in the paper [160], the authors define the (𝜅, 𝛷)-variation of 𝑓 : [0, 1] → ℝ in the natural way by Var𝜅,𝛷 (𝑓; [0, 1]) := sup {

∑∞ 𝑘=1 𝜙𝑘 (|𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| }, ∑∞ 𝑘=1 𝜅(𝑏𝑘 − 𝑎𝑘 )

where the supremum is taken over all collections 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([0, 1]), thus providing a unified approach to the Schramm variation (2.71) and the Koren­ blum variation (2.106). The corresponding space 𝜅𝛷𝐵𝑉([𝑎, 𝑏]) of functions of bounded (𝜅, 𝛷)-variation has been systematically studied afterwards by Park and others in a se­ ries of papers [241–245, 296]. Distortion functions and Korenblum variation occur in entropy theory. In the pa­ per [164], the author defines the 𝜅-entropy of a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) by 𝑚 𝑡𝑗 − 𝑡𝑗−1 ). 𝜅(𝑃) := ∑ 𝜅 ( 𝑏−𝑎 𝑗=1 The 𝜅-entropy of an arbitrary nonempty closed set 𝐹 ⊆ [𝑎, 𝑏] is then defined by 𝜅(𝐹) = 𝜅(𝐹; [𝑎, 𝑏]) := sup {𝜅(𝑃) : 𝑃 ∈ P(𝐹)} , where the supremum is taken over all partitions in the set 𝐹. This entropy has a number of natural properties, see Exercises 2.38 and 2.39. For particular choices of 𝜅, such entropies are well known. Thus, for 𝜅(𝑡) = 𝑡𝛼 , we get the so-called Hölder entropy, for 𝜅(𝑡) = 𝑡(1 − log 𝑡), the Shannon entropy, and for 𝜅(𝑡) = 2/(2 − log 𝑡), the Dini entropy. So-called modular functionals and modular spaces of functions of bounded varia­ tion have been studied by Musielak and Orlicz [235, 236], and later by Leśniewicz and Orlicz [180] and Herda [145]. Theorem 2.74 is due to De la Vallée Poussin [103], see also [272] for a generaliza­ tion to so-called 𝑢-convex functions.²⁸ We point out that Theorem 2.74 carries over to the higher variation Var𝑊 𝑘,1 (𝑓; [𝑎, 𝑏]), see Exercise 2.45 or [273, 274]. Theorem 2.76 was proved in [276] for functions 𝑓 having a bounded second derivative 𝑓󸀠󸀠 on [𝑎, 𝑏], and extended to functions 𝑓 ∈ 𝐴𝐶1 ([𝑎, 𝑏]) in [279]. Of course, the equality (2.150) is a per­ fect analogue to the classical formula 𝑏

Var(𝑓; [𝑎, 𝑏]) = ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 𝑎

which holds for functions 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]); we will come back to this and similar formu­ las in the next chapter.

28 A self-contained account of convex functions and their properties can be found in the monographs [253, 267].

198 | 2 Nonclassical BV-spaces Exercise 2.46 shows that functions from 𝑊𝐵𝑉𝑊 𝑘,1 ([𝑎, 𝑏]) have better smoothness properties if 𝑘 increases. Therefore, the following result [274] is not too surprising. Proposition 2.86. The equalities ∞

∞

⋂ 𝑊𝐵𝑉𝑘,1 ([𝑎, 𝑏]) = 𝐶∞ ([𝑎, 𝑏]) ,

⋃ 𝑊𝐵𝑉𝑘,1 ([𝑎, 𝑏]) = 𝐵𝑉([𝑎, 𝑏])

𝑘=1

𝑘=1

(2.161)

are true. It would be interesting to have a similar explicit description of the classes ∞

∞

𝐴 𝑝 := ⋂ 𝑊𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]) ,

𝐵𝑝 := ⋃ 𝑊𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]) ,

𝑘=1

𝑘=1

see also Problem 2.7 below. In this chapter, we have obtained several results about unions and intersections of various spaces of functions of bounded variation, such as Waterman spaces, Schramm spaces, and related spaces. We collect these results in the following Table 2.8. Table 2.8. Unions and intersections of function classes over 𝐼 = [𝑎, 𝑏]. ∞

⋂ 𝑊𝐵𝑉𝑘,1 (𝐼)

=

𝑘=1 ∞

⋃ 𝑊𝐵𝑉𝑘,1 (𝐼)

=

∩

⋂ Λ𝐵𝑉(𝐼)

=

𝐵𝑉(𝐼)

⋃ 𝑊𝐵𝑉𝑝 (𝐼)

⊂

∩ 𝐻𝐵𝑉(𝐼)

⋃ Λ𝐵𝑉(𝐼)

=

∩ 𝑅(𝐼)

=

∪ 𝐹𝑅(𝐼)

Λ

𝑘=1

𝐶∞ (𝐼)

=

⋂ 𝛷𝐵𝑉(𝐼) 𝛷

𝑝>1

⋃𝜙 𝑊𝐵𝑉𝜙 (𝐼)

=

Λ

⋂ 𝑊𝐵𝑉𝜙 (𝐼)

=

⋃ 𝛷𝐵𝑉(𝐼) 𝛷

𝜙

Given a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), let us recall the Popoviciu variation 𝑚−𝑘+1

Var𝑊 𝑘,1 (𝑓, 𝑃; [𝑎, 𝑏]) = ∑ |𝑓[𝑡𝑗 , . . . , 𝑡𝑗+𝑘−1 ] − 𝑓[𝑡𝑗−1 , . . . , 𝑡𝑗+𝑘−2 ]| 𝑗=1

= |𝑓[𝑡1 , . . . , 𝑡𝑘 ] − 𝑓[𝑡0 , . . . , 𝑡𝑘−1 ]| + |𝑓[𝑡2 , . . . , 𝑡𝑘+1 ] − 𝑓[𝑡1 , . . . , 𝑡𝑘 ]| + . . . + |𝑓[𝑡𝑚−𝑘+1 , . . . , 𝑡𝑚 ] − 𝑓[𝑡𝑚−𝑘 , . . . , 𝑡𝑚−1 ]| defined in (2.142). A similar second variation was defined in [226], where the authors start, for fixed 𝑘 ∈ ℕ, with a “block partition” of the form 𝑄 := {𝜏11 , . . . , 𝜏12𝑘 , 𝜏21 , . . . , 𝜏22𝑘 , . . . , 𝜏𝑛1 , . . . , 𝜏𝑛2𝑘 }

(2.162)

2.7 Comments on Chapter 2 |

199

and associate to this 𝑄 ∈ P([𝑎, 𝑏]) the second variation 𝑛

𝑘+1 , . . . , 𝜏𝑖2𝑘 ] − 𝑓[𝜏𝑖1 , . . . , 𝜏𝑖𝑘 ]| Var𝑀𝑅 𝑘,1 (𝑓, 𝑄; [𝑎, 𝑏]) = ∑ |𝑓[𝜏𝑖 𝑖=1

= |𝑓[𝜏1𝑘+1 , . . . , 𝜏12𝑘 ] − 𝑓[𝜏11 , . . . , 𝜏1𝑘 ]| + |𝑓[𝜏2𝑘+1 , . . . , 𝜏22𝑘 ] − 𝑓[𝜏21 , . . . , 𝜏2𝑘 ]| + . . . + |𝑓[𝜏𝑛𝑘+1 , . . . , 𝜏𝑛2𝑘 ] − 𝑓[𝜏𝑛1 , . . . , 𝜏𝑛𝑘 ]| .

(2.163)

We call the expression 𝑀𝑅 Var𝑀𝑅 𝑘,1 (𝑓; [𝑎, 𝑏]) = sup {Var𝑘,1 (𝑓, 𝑄; [𝑎, 𝑏]) : 𝑄 ∈ P([𝑎, 𝑏])},

(2.164)

where the supremum in (2.164) is taken over all block partitions of the form (2.162), the second variation in the sense of Merentes and Rivas, and denote the corresponding space of all functions 𝑓 satisfying Var𝑀𝑅 𝑘,1 (𝑓; [𝑎, 𝑏]) < ∞ by 𝑀𝑅𝐵𝑉 𝑘,1 ([𝑎, 𝑏]). The follow­ ing proposition shows that in this way, one does not obtain a new space, though they do gain something we already know: 𝑀𝑅 Proposition 2.87. For Var𝑊 𝑘,1 (𝑓; [𝑎, 𝑏]) as in (2.142) and Var𝑘,1 (𝑓; [𝑎, 𝑏]) as in (2.164), the estimates 𝑊 𝑀𝑅 Var𝑀𝑅 𝑘,1 (𝑓; [𝑎, 𝑏]) ≤ Var𝑘,1 (𝑓; [𝑎, 𝑏]) ≤ 2(𝑘 + 1) Var𝑘,1 (𝑓; [𝑎, 𝑏])

(2.165)

are true. Consequently, the spaces 𝑊𝐵𝑉𝑘,1 ([𝑎, 𝑏]) and 𝑀𝑅𝐵𝑉𝑘,1 ([𝑎, 𝑏]) coincide. A sketch of the proof of Proposition 2.87 can be found in [226]. In the following Table 2.9, we compare the definitions of higher order variations in Wiener’s sense introduced in Section 2.7, together with corresponding representation theorems. Table 2.9. Higher order variations and Wiener spaces 𝑊𝐵𝑉𝑘,𝑝 . 𝑊𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏])

𝑝=1

1 0 : Var𝜙 (𝑓/𝜆; [𝑎, 𝑏]) ≤ 1}

is always a nonempty interval.²⁹ More precisely, prove that {[0, ∞] if 𝑓(𝑥) ≡ 0 , 𝐸𝑊 𝜙 (𝑓) = { (‖𝑓‖𝑊𝐵𝑉𝜙 , ∞) if 𝑓(𝑥) ≢ 0 . { Exercise 2.3. Using Exercise 0.75, prove the following analogue to Theorem 1.41: a function 𝑓 belongs to 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) if and only if it may be represented as composi­ tion 𝑓 = 𝑔 ∘ 𝜏, where 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] is increasing and 𝑔 ∈ 𝐿𝑖𝑝𝜙−1 ,∞ ([𝑐, 𝑑]) with 𝑙𝑖𝑝𝜙−1 ,∞ ([𝑐, 𝑑]) ≤ 1, see (0.101). Exercise 2.4. Show that Λ𝐵𝑉([𝑎, 𝑏]) = 𝐵𝑉([𝑎, 𝑏]) if and only if the sequence Λ = (𝜆 𝑛)𝑛 is bounded away from zero, i.e. 𝜆 𝑛 ≥ 𝛿 for all 𝑛 ∈ ℕ and some 𝛿 > 0. Exercise 2.5. Let 𝑍𝐶,𝐷 be the zigzag function (0.91) determined by 𝑐𝑛 :=

1 , 2𝑛

𝑑𝑛 :=

1 𝑛1−𝑞 log(𝑛 + 1)

(𝑛 = 1, 2, 3, . . .) .

Use this function to show that the inclusion in the imbedding (2.48) is strict. Exercise 2.6. Let 𝑍𝜃 be the special zigzag function (0.93) determined by 𝜃 := 1/𝑝. Show that this function satisfies (2.47). Exercise 2.7. By means of Proposition 2.32, Proposition 2.33, Exercise 2.5 and Exer­ cise 2.6, show that ⋃ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊂ Λ 1 𝐵𝑉([𝑎, 𝑏]) = 𝐻𝐵𝑉([𝑎, 𝑏])

𝑝>1

as well as ⋃ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊂ Λ 𝑞0 𝐵𝑉([𝑎, 𝑏]) = 𝑊𝐵𝑉𝑝0 ([𝑎, 𝑏]) ⊂ ⋂ Λ 𝑞 𝐵𝑉([𝑎, 𝑏])

1 0 independent of 𝑛. Exercise 2.20. Show that the modulus of variation (2.52) satisfies the property 𝜈(𝑓)𝑛 ≤ 𝜈(𝑓)𝑚 + 𝜈(𝑓)𝑛−𝑚

(𝑚 = 1, 2, . . . , 𝑛).

Exercise 2.21*. Given a regular function 𝑓 : [𝑎, 𝑏] → ℝ, prove that the condition ∞

∫ log(1 + 𝑁𝑓 (𝑦)) 𝑑𝑦 < ∞ −∞

implies that 𝑓 ∈ 𝐻𝑐 𝐵𝑉([𝑎, 𝑏]). Compare with Proposition 2.31. Exercise 2.22*. Prove the following converse to Proposition 2.47. Suppose that the variation function V𝑓,𝛷 (𝑥) = Var𝛷 (𝑓; [𝑎, 𝑥]) of 𝑓 ∈ 𝛷𝐵𝑉([𝑎, 𝑏]) is continuous at some point 𝑥0 ∈ (𝑎, 𝑏). Show that 𝑓 is then also continuous at 𝑥0 . Exercise 2.23. Prove the following “one-sided version” of Proposition 2.47: if 𝑓 ∈ 𝛷𝐵𝑉([𝑎, 𝑏]) is right continuous at 𝑎 [respectively left continuous at 𝑏], then Var𝛷 (𝑓; [𝑎, 𝑥]) → 0 as 𝑥 → 𝑎+ [respectively Var𝛷 (𝑓; [𝑥, 𝑏]) → 0 as 𝑥 → 𝑏−]. Exercise 2.24. Prove Proposition 2.45. Exercise 2.25. Construct two Schramm sequences 𝛷 = (𝜙𝑛 )𝑛 and 𝛹 = (𝜓𝑛 )𝑛 such that 𝛷𝐵𝑉([0, 1]) ⊆ 𝛹𝐵𝑉([0, 1]), but (2.79) does not hold. Exercise 2.26. For 1 < 𝑝 < 𝑞 < ∞, find a function 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) \ 𝑅𝐵𝑉𝑞 ([𝑎, 𝑏]) using Table 2.4. Exercise 2.27. Show that the oscillation function (0.86) belongs to 𝑅𝐵𝑉𝑝 ([0, 1]) for fixed 𝑝 ∈ [1, ∞) if and only if either 𝛽 > 0 and 𝑝𝛼 + 𝛽 ≥ 𝑝 − 1, or 𝛽 ≤ 0 and 𝑝𝛼 + 𝛽 > 𝑝 − 1. Exercise 2.28. Use Table 2.4 to construct a function 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([0, 1]) \ 𝐿𝑖𝑝([0, 1]). Exercise 2.29. Is the space 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) with norm (2.90) separable? Compare your answer with Exercises 0.6 and 1.49. Exercise 2.30. Define 𝑅𝐵𝑉1𝑝 ([𝑎, 𝑏]) from 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) as in Definition 0.32, equipped with the norm (0.44), i.e. ‖𝑓‖𝑅𝐵𝑉1𝑝 = |𝑓(𝑎)| + ‖𝑓󸀠 ‖𝑅𝐵𝑉𝑝 = |𝑓(𝑎)| + |𝑓󸀠 (𝑎)| + Var𝑅𝑝 (𝑓󸀠 ; [𝑎, 𝑏])1/𝑝 . Show that (𝑅𝐵𝑉1𝑝 ([𝑎, 𝑏]), ‖ ⋅ ‖𝑅𝐵𝑉1𝑝 ) is a Banach space.

2.8 Exercises to Chapter 2

| 205

Exercise 2.31. With the notation of Exercise 2.30, show that 𝑅𝐵𝑉 1𝑝 ([𝑎, 𝑏]) 󳨅→ 𝐿𝑖𝑝([𝑎, 𝑏]) for 𝑝 > 1, and calculate the sharp imbedding constant 𝑐(𝑅𝐵𝑉1𝑝 , 𝐿𝑖𝑝). Exercise 2.32. For fixed 𝛼 ∈ (0, 1), let 𝑓 : [0, 1] → ℝ be defined by {2𝑥 𝑓(𝑥) := { (2𝑏 − 2)𝑥 + 2 − 𝑏𝛼 { 𝛼

for 0 ≤ 𝑥 ≤ for

1 2

1 2

,

< 𝑥 ≤ 1,

where 𝑏𝛼 := 2/(1 + 21−𝛼 ). Since 21−𝛼 > 1, the function 𝑓 is increasing on [0, 1/2] with 𝑓(0) = 0 and 𝑓(1/2) = 1, and decreasing on [1/2, 1] with 𝑓(1) = 𝑏𝛼 < 1. (a) Show that 𝑓 ∈ 𝜅𝐵𝑉([0, 1]), where 𝜅(𝑡) = 𝑡𝛼 . (b) Calculate Var𝜅 (𝑓; [0, 1]) by only considering the special partitions 𝑃1 := {0, 1} and 𝑃2 := {0, 1/2, 1} of [0, 1]. (c) Prove that the variation function V𝑓,𝜅 defined in (2.115) coincides with 𝑓 and is therefore not monotonically increasing. Exercise 2.33. Prove Proposition 2.64 without the additional assumption that 𝑓(0) = 𝑓(1). Exercise 2.34*. Given a distortion function 𝜅 : [0, 1] → [0, 1], construct a function 𝑓 ∈ 𝑅([0, 1]) \ 𝜅𝐵𝑉([0, 1]). Can you also construct a function 𝑓 ∈ 𝑅([0, 1]) \ (⋃ 𝜅𝐵𝑉([0, 1])) , 𝜅

where the union is taken over all distortion functions 𝜅? Exercise 2.35. Given an arbitrary distortion function 𝜅 : [0, 1] → [0, 1], imitate the construction in Example 2.66 to find a function 𝑓 ∈ 𝜅𝐵𝑉([0, 1]) \ 𝐵𝑉([0, 1]). Exercise 2.36. Combine Theorem 2.67 with Theorem 2.68 to derive and prove a Hellytype selection theorem for sequences of functions of bounded Korenblum variation. Exercise 2.37. Given an (𝑛 + 2)-tuple (𝑎0 , 𝑎1 , . . . , 𝑎𝑛 , 𝑎𝑛+1 ) ∈ ℝ𝑛+2 with 𝑎0 = 𝑎𝑛+1 = 0, show that 𝑛 1 𝑛+1 ∑(𝑎𝑖 − 𝑎𝑖−1 )+ + 𝑎𝑛− = ∑ |𝑎𝑖 − 𝑎𝑖−1 | , 2 𝑖=1 𝑖=1 where 𝑎+ and 𝑎− are defined by (2.113). Use this to carry out the details of the first step in the proof of Theorem 2.68. Exercise 2.38. Show that the 𝜅-entropy 𝜅(𝑃) defined in Section 2.8 has the follow­ ing property: if 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), then 𝜅(𝑃) ≤ 𝑚/𝜅(1/𝑚). Moreover, this estimate is sharp and attained for equidistant partitions 𝑃. Exercise 2.39. Show that the 𝜅-entropy 𝜅(𝐹) defined in Section 2.8 satisfies 𝜅(𝐹) ≤ 𝜅(𝐺) for 𝐹 ⊆ 𝐺 as well as 𝜅(𝐹 ∪ 𝐺) ≤ 𝜅(𝐹) + 𝜅(𝐺) for arbitrary closed sets 𝐹 and 𝐺.

206 | 2 Nonclassical BV-spaces Exercise 2.40. In analogy to Exercise 1.56, we say that a function 𝑓 ∈ 𝜅𝐵𝑉([𝑎, 𝑏]) has vanishing 𝜅-variation at 𝑡0 ∈ [𝑎, 𝑏] if lim Var𝜅 (𝐹𝑡0 ,𝛿 ; [𝑎, 𝑏]) = 0 ,

𝛿→0

where 𝐹𝑡0 ,𝛿 is defined as in Exercise 1.56. Show that, in contrast to 𝐵𝑉-functions, a function 𝑓 ∈ 𝜅𝐵𝑉 which is continuous at 𝑡0 does not necessarily have vanishing 𝜅variation at 𝑡0 . Exercise 2.41. For points 𝑡0 , 𝑡1 , . . . , 𝑡𝑘 ∈ [𝑎, 𝑏] and indices 𝑗 = 0, 1, . . . , 𝑘, let 𝑃𝑗 (𝑡0 , . . . , 𝑡𝑘 ) := ∏(𝑡𝑖 − 𝑡𝑗 ) = (𝑡0 − 𝑡𝑗 ) ⋅ ⋅ ⋅ (𝑡𝑗−1 − 𝑡𝑗 )(𝑡𝑗+1 − 𝑡𝑗 ) ⋅ ⋅ ⋅ (𝑡𝑘 − 𝑡𝑗 ) . 𝑖=𝑗̸

Prove by induction that 𝑘

𝑓[𝑡0 , 𝑡1 , . . . , 𝑡𝑘 ] = ∑

𝑓(𝑡𝑗 )

𝑃 (𝑡 , . . . , 𝑡𝑘 ) 𝑗=0 𝑗 0

,

where 𝑓[𝑡0 , 𝑡1 , . . . , 𝑡𝑘 ] is given by (2.139). Exercise 2.42. Let 𝑓 ∈ 𝐶𝑘 ([𝑎, 𝑏]) and 𝑎 < 𝑡0 < 𝑏. Show that lim 𝑓[𝑡⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 0 , . . . , 𝑡0 , 𝑡0 + ℎ] = ℎ→0 𝑘 times

𝑓(𝑘) (𝑡0 ) , 𝑘!

where 𝑓[𝑡0 , 𝑡1 , . . . , 𝑡𝑘 ] is given by (2.139). Exercise 2.43. Prove the following weaker form of Exercise 2.42. Let 𝑓 ∈ 𝐶𝑘−1 ([𝑎, 𝑏]) and 𝑎 < 𝑡0 < 𝑏. Suppose that the unilateral derivatives 𝑓−(𝑘) and 𝑓+(𝑘) exist everywhere in [𝑎, 𝑏], and the usual derivative 𝑓(𝑘) exists in [𝑎, 𝑏] \ 𝑁, where 𝑁 is countable. Show that 𝑓−(𝑘) (𝑡0 ) lim 𝑓[𝑡⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ , . . . , 𝑡 , 𝑡 + ℎ] = 0 0 0 ℎ→0− 𝑘! 𝑘 times

and lim 𝑓[𝑡⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 0 , . . . , 𝑡0 , 𝑡0 + ℎ] = ℎ→0+ 𝑘 times

𝑓+(𝑘) (𝑡0 ) , 𝑘!

where 𝑓[𝑡0 , 𝑡1 , . . . , 𝑡𝑘 ] is given by (2.139). Exercise 2.44*. Prove that every function 𝑓 ∈ 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]) is Lipschitz continuous. Exercise 2.45*. Prove the following higher order analogue of Theorem 2.74: a func­ tion 𝑓 : [𝑎, 𝑏] → ℝ has bounded 𝑘-th variation Var𝑊 𝑘,1 (𝑓; [𝑎, 𝑏]) if and only if its derivative (𝑘−2) 𝑓 may be represented as a difference of two convex functions. Exercise 2.46. Prove that every function 𝑓 ∈ 𝑊𝐵𝑉𝑘,1 ([𝑎, 𝑏]) admits unilateral deriva­ tives 𝑓+(𝑘−1) and 𝑓−(𝑘−1) on [𝑎, 𝑏]. Moreover, show that there exists a nullset 𝑁 ⊂ [𝑎, 𝑏] such that 𝑓 ∈ 𝐶𝑘−1 ([𝑎, 𝑏] \ 𝑁).

2.8 Exercises to Chapter 2

| 207

Exercise 2.47. Show that the inclusions (2.144) and (2.145) are strict. Exercise 2.48. Show by means of the function 𝑓 : [0, 2] → ℝ defined by 𝑓(𝑥) := max {0, 𝑥 − 1} that the estimate (2.146) is sharp. Exercise 2.49. Suppose that 𝑓 : [𝑎, 𝑏] → ℝ is 𝑘-times differentiable, and 𝑓(𝑘) is Rie­ mann integrable over [𝑎, 𝑏]. Prove that 𝑏 𝑡1

𝑡𝑘−1

𝑓[𝑡0 , 𝑡1 , . . . , 𝑡𝑘 ] = ∫ ∫ ⋅ ∫ 𝑓(𝑘) (𝜏𝑘 (𝑡𝑘 − 𝑡𝑘−1 ) + . . . + 𝜏1 (𝑡1 − 𝑡0 ) + 𝑡0 ) 𝑑𝜏𝑘 . . . 𝑑𝜏1 , 𝑎 0

0

where 𝑓[𝑡0 , 𝑡1 , . . . , 𝑡𝑘 ] is given by (2.139). Exercise 2.50. Show that Theorem 2.77 also holds true if we replace, in (2.151), the space 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) by the class V𝑊 𝜙 ([𝑎, 𝑏]) introduced in Definition 2.2. Exercise 2.51. Prove Proposition 2.79. Exercise 2.52*. Prove the “only if” part in Proposition 2.80. Exercise 2.53. Using Exercise 0.75, prove Proposition 2.81.

3 Absolutely continuous functions Absolutely continuous functions are intimately related to functions of bounded varia­ tion in several respects. First, absolute continuity is equivalent to the combination of three properties, namely, continuity, bounded variation, and invariance of nullsets; this is the assertion of the Vitali–Banach–Zaretskij theorem which we will prove in Section 3.2. Second, the absolutely continuous functions on [𝑎, 𝑏] are precisely those functions 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) whose derivatives 𝑓󸀠 (which exist a.e. by Lebesgue’s differen­ tiation theorem) belong to 𝐿 1 ([𝑎, 𝑏]); moreover, the fundamental theorem of calculus (in the Lebesgue integral version) holds in this case. This shows that the definition of absolute continuity is both important and natural. If one replaces the condition 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) with the stronger condition 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) (1 < 𝑝 < ∞) in this statement, one gets precisely the condition 𝑓󸀠 ∈ 𝐿 𝑝 ([𝑎, 𝑏]); this is the statement of the famous Riesz theorem which we will discuss, together with some generalization due to Yu. T. Medvedev, in Section 3.5. Section 3.4 is concerned with some relations between bounded variation, absolute continuity, and functions with rectifiable graphs.

3.1 Continuity and absolute continuity Although we have introduced the concept of absolute continuity in Definition 1.21, for the reader’s ease, we repeat the definition here. Definition 3.1. Given a compact interval [𝑎, 𝑏], by 𝛴([𝑎, 𝑏]), we denote the family of all finite systems 𝑆 := {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛]} (𝑛 ∈ ℕ variable) of pairwise nonoverlapping subintervals [𝑎𝑘 , 𝑏𝑘 ] ⊆ [𝑎, 𝑏]. For 𝑆 ∈ 𝛴([𝑎, 𝑏]) and 𝑓 : [𝑎, 𝑏] → ℝ, we put 𝑛

𝛩(𝑆) := ∑ |𝑏𝑘 − 𝑎𝑘 |, 𝑘=1

𝑛

𝛤(𝑓; 𝑆) := ∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| .

(3.1)

𝑘=1

We call the function 𝑓 absolutely continuous on [𝑎, 𝑏] and write 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) if for each 𝜀 > 0, there exists some 𝛿 > 0 such that 𝛤(𝑓; 𝑆) ≤ 𝜀 for any 𝑆 ∈ 𝛴([𝑎, 𝑏]) satisfying 𝛩(𝑆) ≤ 𝛿. ◼ Since the number 𝑛 in (3.1) is arbitrary, we could replace the finite sums in Definition 3.1 by infinite series, see Exercise 3.7. We point out that it is very important to only choose mutually nonoverlapping subintervals [𝑎𝑘 , 𝑏𝑘 ] ⊆ [𝑎, 𝑏]. If we drop this assumption, we get an essentially smaller class of functions, see Exercise 3.8. For further use, we collect some natural algebraic properties of the function class 𝐴𝐶([𝑎, 𝑏]). Proposition 3.2. Let 𝑓, 𝑔 ∈ 𝐴𝐶([𝑎, 𝑏]) and 𝜇 ∈ ℝ. Then 𝑓 + 𝑔, 𝜇𝑓 and 𝑓𝑔 also belong to 𝐴𝐶([𝑎, 𝑏]). Moreover, in case 𝑔(𝑥) ≠ 0 for 𝑎 ≤ 𝑥 ≤ 𝑏, we also have 𝑓/𝑔 ∈ 𝐴𝐶([𝑎, 𝑏]).

3.1 Continuity and absolute continuity

| 209

Proof. Let 𝜀 > 0, and choose 𝛿 > 0 such that both 𝛤(𝑓; 𝑆) ≤ 𝜀 and 𝛤(𝑔; 𝑆) ≤ 𝜀 for any 𝑆 ∈ 𝛴([𝑎, 𝑏]) satisfying 𝛩(𝑆) ≤ 𝛿. Then 𝛤(𝑓 + 𝑔; 𝑆) ≤ 𝛤(𝑓; 𝑆) + 𝛤(𝑔; 𝑆) ≤ 2𝜀 , which shows that 𝑓+𝑔 ∈ 𝐴𝐶([𝑎, 𝑏]). Proving that 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) implies 𝜇𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) goes similarly by using the fact that 𝛤(𝜇𝑓; 𝑆) = |𝜇|𝛤(𝑓; 𝑆) for all 𝜇 ∈ ℝ and 𝑆 ∈ 𝛴([𝑎, 𝑏]). Since every absolutely continuous function is continuous, we know that ‖𝑓‖𝐶 ≤ 𝑐 and ‖𝑔‖𝐶 ≤ 𝑐 for some 𝑐 > 0, where ‖ ⋅ ‖𝐶 denotes the norm (0.45). Given 𝜀 > 0, choose again 𝛿 > 0 such that both 𝛤(𝑓; 𝑆) ≤ 𝜀 and 𝛤(𝑔; 𝑆) ≤ 𝜀 for any 𝑆 ∈ 𝛴([𝑎, 𝑏]) satisfying 𝛩(𝑆) ≤ 𝛿. For any subinterval [𝑎𝑘 , 𝑏𝑘 ] ⊆ [𝑎, 𝑏], we then have |𝑓(𝑎𝑘 )𝑔(𝑎𝑘 ) − 𝑓(𝑏𝑘 )𝑔(𝑏𝑘 )| ≤ |𝑓(𝑎𝑘 )| |𝑔(𝑎𝑘 ) − 𝑔(𝑏𝑘 )| + |𝑓(𝑎𝑘 ) − 𝑓(𝑏𝑘 )| |𝑔(𝑏𝑘 )| ≤ 𝑐 |𝑔(𝑎𝑘 ) − 𝑔(𝑏𝑘 )| + 𝑐 |𝑓(𝑎𝑘 ) − 𝑓(𝑏𝑘 )| , and hence 𝛤(𝑓𝑔; 𝑆) ≤ 𝑐(𝛤(𝑓; 𝑆) + 𝛤(𝑔; 𝑆)) ≤ 2𝑐𝜀 for all 𝑆 ∈ 𝛴([𝑎, 𝑏]) satisfying 𝛩(𝑆) ≤ 𝛿. Thus, we have proved that 𝑓𝑔 ∈ 𝐴𝐶([𝑎, 𝑏]). It remains to prove that 1/𝑔 ∈ 𝐴𝐶([𝑎, 𝑏]) for all 𝑔 ∈ 𝐴𝐶([𝑎, 𝑏]) satisfying 𝑔(𝑥) ≠ 0 on [𝑎, 𝑏]. Given 𝜀 > 0, choose 𝛿 > 0 such that 𝛤(𝑔; 𝑆) ≤ 𝜀 for any 𝑆 ∈ 𝛴([𝑎, 𝑏]) satisfying 𝛩(𝑆) ≤ 𝛿. Again, from the continuity of 𝑔, we deduce that |𝑔(𝑥)| ≥ 𝑐 on [𝑎, 𝑏] for some 𝑐 > 0. Thus, for any subinterval [𝑎𝑘 , 𝑏𝑘 ] ∈ 𝑆, we have 󵄨󵄨 1 1 󵄨󵄨󵄨󵄨 |𝑔(𝑎𝑘 ) − 𝑔(𝑏𝑘 )| |𝑔(𝑎𝑘 ) − 𝑔(𝑏𝑘 )| 󵄨󵄨 − ≤ , 󵄨󵄨 = 󵄨󵄨 |𝑔(𝑎𝑘 )𝑔(𝑏𝑘 )| 𝑐2 󵄨󵄨 𝑔(𝑎𝑘 ) 𝑔(𝑏𝑘 ) 󵄨󵄨 and hence 𝛤(1/𝑔; 𝑆) ≤

𝛤(𝑔; 𝑆) 𝜀 ≤ 2 𝑐2 𝑐

which proves the assertion. It is again illuminating to compare the conditions 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) and |𝑓| ∈ 𝐴𝐶([𝑎, 𝑏]) (Exercises 3.1–3.3), as we have done for the function space 𝐵𝑉([𝑎, 𝑏]) in Exercises 1.3–1.5 in Chapter 1. The following elementary result on continuous functions is taught in every first-year calculus course: if 𝑓 : [𝑎, 𝑏] → [𝑓(𝑎), 𝑓(𝑏)] is continuous and strictly in­ creasing, then 𝑓−1 : [𝑓(𝑎), 𝑓(𝑏)] → [𝑎, 𝑏] exists and is also continuous and strictly increasing, and so 𝑓 is a homeomorphism. A corresponding result may be stated for absolutely continuous functions, but it is more delicate, see Exercise 3.11. Since every function 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) has bounded variation, by (1.46), we may con­ sider its variation function 𝑉𝑓 defined in (1.13). Theorem 1.26 (e) shows that the abso­ lute continuity of 𝑓 carries over to 𝑉𝑓 ; this immediately implies the following Proposition 3.3. Every function 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) may be represented as a difference of two absolutely continuous increasing functions.

210 | 3 Absolutely continuous functions Since 𝐴𝐶([𝑎, 𝑏]) is a linear subspace of 𝐵𝑉([𝑎, 𝑏]), the question arises if this subspace is closed in 𝐵𝑉([𝑎, 𝑏]) in the norm (1.16), i.e. if (𝐴𝐶([𝑎, 𝑏]), ‖ ⋅ ‖𝐵𝑉 ) is a Banach space. This is in fact true, as we will show in the next section (Proposition 3.24). The next proposition shows that, roughly speaking,“indefinite integrals” of 𝐿 1 functions are always absolutely continuous. Proposition 3.4. Let 𝑔 ∈ 𝐿 1 ([𝑎, 𝑏]) and 𝑥

𝑓(𝑥) := ∫ 𝑔(𝑡) 𝑑𝑡

(𝑎 ≤ 𝑥 ≤ 𝑏) .

(3.2)

𝑎

Then 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]). Proof. Let 𝜀 > 0. From the absolute continuity of the Lebesgue integral (Theorem 0.6), it follows that there exists a 𝛿 > 0 such that ∫ |𝑔(𝑥)| 𝑑𝑥 ≤ 𝜀 𝑀

for all subsets 𝑀 ⊆ [𝑎, 𝑏] with 𝜆(𝑀) ≤ 𝛿. Consider 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛]} ∈ 𝛴([𝑎, 𝑏]) with 𝛩(𝑆) ≤ 𝛿, and put 𝑀 := [𝑎1 , 𝑏1 ] ∪ . . . ∪ [𝑎𝑛 , 𝑏𝑛 ]. Then 󵄨󵄨 𝑏𝑘 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 𝛤(𝑓; 𝑆) = ∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| = ∑ 󵄨󵄨󵄨∫ 𝑔(𝑡) 𝑑𝑡󵄨󵄨󵄨󵄨 ≤ ∫ |𝑔(𝑥)| 𝑑𝑥 ≤ 𝜀 󵄨 󵄨󵄨 𝑘=1 𝑘=1 󵄨󵄨𝑎 󵄨󵄨 𝑀 󵄨𝑘 𝑛

𝑛

since 𝜆(𝑀) ≤ 𝛩(𝑆) ≤ 𝛿, by construction. Recall that the analogue to Proposition 3.4 for the Riemann integral reads as follows: if 𝑔 : [𝑎, 𝑏] → ℝ is Riemann integrable and bounded, the function 𝑓 defined in (3.2) is Lipschitz continuous. A certain converse of Proposition 3.4 will be proved in the following section (Theo­ rem 3.18). Here, we use Proposition 3.4 to give an example of an absolutely continuous function which is not Hölder continuous for any 𝛼 ∈ (0, 1]. Example 3.5. Let 𝑓 : [0, 1] → ℝ be the function from Example 1.25 (or 0.41). As we have seen there, 𝑓 ∈ ̸ 𝐿𝑖𝑝𝛼 ([0, 1]) for any 𝛼 > 0. Now, the derivative 𝑓󸀠 = 𝑔 exists everywhere and has the form 2 for 0 < 𝑥 ≤ 1 , { 2 2 𝑔(𝑥) = { 𝑥 log 𝑥 for 𝑥 = 0 . {0 Since the (improper) Riemann integral of 𝑔 over (0, 1) exists and 𝑔(𝑥) > 0 on (0, 1), we conclude that 𝑔 ∈ 𝐿 1 ([0, 1]). Thus, from Proposition 3.4, it follows that 𝑓 satisfies (3.2) and is absolutely continuous on [0, 1]. ♥ Later (Theorem 3.19), we will show that not only an absolutely continuous function, but also its total variation may be expressed through an integral.

3.2 The Vitali–Banach–Zaretskij theorem | 211

3.2 The Vitali–Banach–Zaretskij theorem As we have seen, any absolutely continuous function is both continuous and of bounded variation. We now show by means of a famous example that the converse is not true. Recall that the Cantor set 𝐶 ⊂ [0, 1] may be defined as intersection ∞

(3.3)

𝐶 := ⋂ 𝐶𝑛 , 𝑛=0

where 2 2 3 6 7 8 1 1 𝐶1 := [0, ] ∪ [ , 1] , 𝐶2 := [0, ] ∪ [ , ] ∪ [ , ] ∪ [ , 1] , . . . 3 3 9 9 9 9 9 9 𝐶𝑛 := [0, 3−𝑛 ] ∪ [2 ⋅ 3−𝑛 , 3 ⋅ 3−𝑛 ] ∪ . . . ∪ [(3𝑛 − 1) ⋅ 3−𝑛 , 1] . (3.4) 𝐶0 := [0, 1] ,

Alternatively, 𝐶 may be defined as a set of all elements 𝑥 ∈ [0, 1] whose ternary representation¹ ∞

𝑥 = 0.𝑥1 𝑥2 𝑥3 . . . = ∑ 𝑥𝑘 3−𝑘

(𝑥𝑘 ∈ {0, 1, 2})

(3.5)

𝑘=1

only contains the numbers 0 and 2. This set has many interesting properties: for in­ stance, it is a perfect, compact and uncountable nullset.² Furthermore, the function 𝜑 : 𝐶 → [0, 1] defined by ∞ ∞ 𝑥 (3.6) 𝜑 ( ∑ 𝑥𝑛 3−𝑛 ) = ∑ 𝑛 2−𝑛 , 𝑛=1 𝑛=1 2 where the element on the right-hand side of (3.6) is represented in the binary system, is monotonically increasing and surjective, i.e. satisfies 𝜑(𝐶) = [0, 1]. Since the function 𝜑 takes the same values at the endpoints of two adjacent intervals in (3.4), 𝜑 admits an increasing continuous extension from 𝐶 to the whole interval [0, 1] which we still denote by 𝜑. This function will be called the Cantor function in what follows.³ The map 𝜓 : [0, 1] → [0, 1] defined by 𝜓(𝑥) :=

1 (𝑥 + 𝜑(𝑥)) 2

(3.7)

which we call strict Cantor function is also of interest. Since 𝜓 is, in contrast to 𝜑, strictly increasing, it is even a homeomorphism of [0, 1] onto itself. These functions have the required property which we announced before:

1 Observe that the representation (3.5) is not unique: for example, 13 = 0.100000 . . . = 0.022222 . . .; however, for our purposes, this is not relevant. 2 Recall that a set is perfect if it only consists of accumulation points. 3 Since the continuous extension of 𝜑 from 𝐶 to [0, 1] is constant on [0, 1] \ 𝐶, it is even differentiable a.e. on [0, 1] because 𝐶 is a nullset; more precisely, 𝜑󸀠 (𝑥) ≡ 0 on [0, 1] \ 𝐶.

212 | 3 Absolutely continuous functions Example 3.6. Being monotone and continuous, the Cantor function 𝜑 : [0, 1] → [0, 1] belongs to 𝐵𝑉([0, 1]) ∩ 𝐶([0, 1]). However, 𝜑 is not absolutely continuous on [0, 1]. To see this, we choose, as intervals in (3.1), precisely the 2𝑛 disjoint intervals of length 3−𝑛 remaining in the 𝑛-th construction step of the Cantor set 𝐶, i.e. the intervals occurring in the union 𝐶𝑛 in (3.4). Collecting these intervals (in their natural order) in the family 𝑆 := {[𝑎1 , 𝑏1 ], . . . , [𝑎2𝑛 , 𝑏2𝑛 ]}, we get, on the one hand, 2𝑛

𝛩(𝑆) = ∑ |𝑏𝑘 − 𝑎𝑘 | = 1 − 𝑘=1

4 2𝑛−1 1 2 2 𝑛 − − − ... − 𝑛 = ( ) , 3 9 27 3 3

and this may be made arbitrarily small by choosing 𝑛 sufficiently large. On the other hand, 2𝑛

𝛤(𝑓; 𝑆) = ∑ |𝜑(𝑏𝑘 ) − 𝜑(𝑎𝑘 )| = 𝜑(1) − 𝜑(0) = 1 𝑘=1

since by construction, all terms cancel out, except for the first and the last one. This shows that 𝜑 ∈ ̸ 𝐴𝐶([0, 1]) as claimed. Of course, the strict Cantor function 𝜓 defined in (3.7) may also serve as an example. ♥ In a moment, we will give an alternative and rather elegant proof of the fact that the function (3.6) is not absolutely continuous, see Theorem 3.9. Functions like the Cantor function 𝜑 discussed in Example 3.6 are so important in the theory of real functions that they have a special name: Definition 3.7. A nonconstant function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]) is called singular if it is differentiable a.e. on [𝑎, 𝑏] with 𝑓󸀠 (𝑥) = 0. ◼ In view of Example 3.6, the question arises as to what property is “missing” if a func­ tion is continuous and of bounded variation, but not absolutely continuous, as the Cantor functions (3.6) and (3.7). Here, the following definition is crucial. Definition 3.8. We say that a function 𝑓 : [𝑎, 𝑏] → ℝ satisfies the Luzin condition⁴ if 𝑓 maps every nullset into a nullset. We denote the set of all functions satisfying the Luzin condition on [𝑎, 𝑏] by 𝐿𝑢([𝑎, 𝑏]). ◼ The most prominent example of a function which fails to satisfy the Luzin condition is the Cantor function (3.6). Indeed, 𝐶 is a nullset, but 𝜑(𝐶) = [0, 1] is not. Using the Luzin condition, we are now in the position to formulate and prove the main result of this section which is usually referred to as the Vitali–Banach–Zaretskij theorem.

4 This condition was introduced by the Soviet mathematician N. N. Luzin; in the literature, it is some­ times called the property (N), where N stands for “nullset.”

3.2 The Vitali–Banach–Zaretskij theorem |

213

Theorem 3.9 (Vitali–Banach–Zaretskij). A function 𝑓 : [𝑎, 𝑏] → ℝ is absolutely con­ tinuous if and only if 𝑓 is continuous, has bounded variation, and satisfies the Luzin condition. Proof. To prove the “if” part, we suppose that 𝑓 is continuous, has bounded variation, and satisfies the Luzin condition; we have to show that 𝑓 is absolutely continuous. Suppose that 𝑓 ∈ ̸ 𝐴𝐶([𝑎, 𝑏]). Then there exists 𝜀0 > 0 such that for every 𝑛 ∈ ℕ, we find a collection 𝑆𝑛 := {[𝑎1,𝑛 , 𝑏1,𝑛 ], [𝑎2,𝑛 , 𝑏2,𝑛 ], . . . , [𝑎𝑘𝑛 ,𝑛 , 𝑏𝑘𝑛,𝑛 ]} ∈ 𝛴([𝑎, 𝑏]) (𝑘𝑛 ∈ ℕ) with 𝛩(𝑆𝑛 ) ≤ 1/𝑛2 and 𝛤(𝑓; 𝑆𝑛) > 𝜀0 . Let 𝑚𝑖,𝑛 := inf {𝑓(𝑥) : 𝑎𝑖,𝑛 ≤ 𝑥 ≤ 𝑏𝑖,𝑛 } ,

(3.8)

𝑀𝑖,𝑛 := sup {𝑓(𝑥) : 𝑎𝑖,𝑛 ≤ 𝑥 ≤ 𝑏𝑖,𝑛 }

(3.9)

and

𝑘𝑛

∞

𝐸𝑛 := ⋃(𝑎𝑖,𝑛 , 𝑏𝑖,𝑛 ), 𝑖=1

∞

𝑁 := ⋂ ⋃ 𝐸𝑛 .

(3.10)

𝑚=1 𝑛=𝑚

Then 𝑁 is a nullset, and thus also 𝑓(𝑁) since 𝑓 satisfies the Luzin condition. In addition to (3.8) and (3.9), we need the numbers 𝑚(𝑓) and 𝑀(𝑓) defined in (0.61) and (0.62), respectively. For 𝑛 = 1, 2, 3, . . . and 𝑖 = 1, 2, . . . , 𝑘𝑛 , consider the functions 𝑔𝑖,𝑛 : ℝ → {0, 1} and 𝑔𝑛 : ℝ → ℕ0 defined by {1 if 𝑓(𝑥) = 𝑦 for some 𝑥 ∈ (𝑎𝑖,𝑛 , 𝑏𝑖,𝑛 ) , 𝑔𝑖,𝑛 (𝑦) := { 0 otherwise { and 𝑔𝑛 (𝑦) := 𝑔1,𝑛 (𝑦) + 𝑔2,𝑛 (𝑦) + . . . + 𝑔𝑘𝑛 ,𝑛 (𝑦) , respectively. Then we have ∞

𝑘𝑛

𝑘𝑛

𝑖=1

𝑖=1

󵄨 󵄨 ∫ 𝑔𝑛 (𝑦) 𝑑𝑦 = ∑ (𝑀𝑖,𝑛 − 𝑚𝑖,𝑛) ≥ ∑ 󵄨󵄨󵄨𝑓(𝑏𝑖,𝑛 ) − 𝑓(𝑎𝑖,𝑛 )󵄨󵄨󵄨 > 𝜀0 . −∞

(3.11)

Moreover, 𝑔𝑛 (𝑦) ≤ 𝐼𝑓 (𝑦) for all 𝑛, where 𝐼𝑓 : [𝑚(𝑓), 𝑀(𝑓)] → ℝ denotes the Banach indicatrix of 𝑓 (Definition 0.38). Consider the sets 𝑁∞ (𝑓) := {𝑦 ∈ ℝ : 𝐼𝑓 (𝑦) = ∞},

𝑁0 (𝑓) := {𝑦 ∈ ℝ : 𝑔𝑛 (𝑦) ↛ 0 (𝑛 → ∞)} .

Since 𝑓 ∈ 𝐶([𝑎, 𝑏]) ∩ 𝐵𝑉([𝑎, 𝑏]), we know that 𝐼𝑓 ∈ 𝐿 1 (ℝ), see Proposition 1.27, and so 𝑁∞ (𝑓) is a nullset. Let 𝑦 ∈ 𝑁0 \ 𝑁∞ ; in particular, 𝐼𝑓 (𝑦) < ∞. Then we find a subsequence (𝑔𝑛𝑘 )𝑘 of (𝑔𝑛 )𝑛 such that 𝑔𝑛𝑘 (𝑦) ≥ 1 for all 𝑘. For each 𝑘, choose 𝑥𝑘 ∈ 𝐸𝑛𝑘 with 𝑓(𝑥𝑘 ) = 𝑦. Since the set 𝑓−1 (𝑦) ∩ [𝑎, 𝑏] is finite, we find at least one point 𝑥0 in this set which belongs to infinitely many of the sets 𝐸𝑛𝑘 , and so 𝑥0 ∈ 𝑁, see (3.10) and 𝑦 = 𝑓(𝑥0 ) ∈ 𝑓(𝑁). However, we have already seen above that 𝑓(𝑁) is a nullset, and so 𝑁0 (𝑓) ⊆ 𝑓(𝑁) ∪ 𝑁∞ (𝑓) is also a nullset. By definition of 𝑁0 (𝑓), this means

214 | 3 Absolutely continuous functions that 𝑔𝑛 (𝑦) → 0, as 𝑛 → ∞, for almost all 𝑦 ∈ [𝑚(𝑓), 𝑀(𝑓)]. Since 𝑔𝑛 (𝑦) ≤ 𝐼𝑓 (𝑦) and 𝐼𝑓 ∈ 𝐿 1 (ℝ), from Lebesgue’s dominated convergence theorem (Theorem 0.4) we conclude that ∞

lim ∫ 𝑔𝑛 (𝑦) 𝑑𝑦 = 0 ,

𝑛→∞

−∞

contradicting (3.11). This contradiction shows that our assumption was false, and so 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]). Now, we prove the converse implication, i.e. the “only if” part. We already know from Proposition 1.22 that the inclusion 𝐴𝐶([𝑎, 𝑏]) ⊆ 𝐶([𝑎, 𝑏]) ∩ 𝐵𝑉([𝑎, 𝑏]) is true. Consequently, to prove the “only if” part, we merely have to show that an ab­ solutely continuous function satisfies the Luzin condition. Therefore, let 𝑁 ⊂ [𝑎, 𝑏] be a nullset and 𝜀 > 0. Exercise 3.7 shows that we may find a 𝛿 > 0 such that whenever {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]) is a sequence of nonoverlapping subintervals of [𝑎, 𝑏], the condition ∞

∑ |𝑏𝑘 − 𝑎𝑘 | ≤ 𝛿 𝑘=1

implies the condition

∞

∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| ≤ 𝜀 . 𝑘=1

Since 𝑁 is a nullset, we may further choose an open set 𝐺 ⊃ 𝑁 with 𝜆(𝐺) ≤ 𝛿. Now, we use the well-known fact that being an open set, 𝐺 may be represented as a countable disjoint union of open intervals (𝑎1 , 𝑏1 ), (𝑎2 , 𝑏2 ), (𝑎3 , 𝑏3 ), . . . and obtain ∞

∞

∑ |𝑏𝑘 − 𝑎𝑘 | = 𝜆 (⋃(𝑎𝑘 , 𝑏𝑘 )) = 𝜆(𝐺) ≤ 𝛿 . 𝑘=1

𝑘=1

For each 𝑘, we choose points⁵ 𝑥𝑘 and 𝑦𝑘 with 𝑓([𝑎𝑘 , 𝑏𝑘 ]) = [𝑓(𝑥𝑘 ), 𝑓(𝑦𝑘 )], and let 𝑚𝑘 := min {𝑥𝑘 , 𝑦𝑘 } and 𝑀𝑘 := max {𝑥𝑘 , 𝑦𝑘 }. Then the intervals [𝑚𝑘 , 𝑀𝑘 ] are pairwise nonoverlapping and satisfy ∞

∞

∞

∑ |𝑥𝑘 − 𝑦𝑘 | = ∑ (𝑀𝑘 − 𝑚𝑘 ) ≤ ∑ |𝑏𝑘 − 𝑎𝑘 | ≤ 𝛿 . 𝑘=1

𝑘=1

𝑘=1

However, ∞

𝑓(𝑁) ⊆ ⋃[𝑓(𝑥𝑘 ), 𝑓(𝑦𝑘 )], 𝑘=1

∞

∑ |𝑓(𝑥𝑘 ) − 𝑓(𝑦𝑘 )| ≤ 𝜀 𝑘=1

since 𝑓 is absolutely continuous, by assumption. Since 𝜀 > 0 was arbitrary, we con­ clude that 𝑓(𝑁) is a nullset, and so the proof is complete.

5 Here, we use the continuity of 𝑓, so 𝑓 attains its supremum and infimum on each compact interval.

3.2 The Vitali–Banach–Zaretskij theorem |

215

The statement of Theorem 3.9 may be summarized as the equality of sets 𝐴𝐶([𝑎, 𝑏]) = 𝐶([𝑎, 𝑏]) ∩ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐿𝑢([𝑎, 𝑏]) .

(3.12)

Theorem 3.9 explains again why the function 𝑓 from Example 3.5 is absolutely con­ tinuous: in Example 1.25, we have seen that 𝑓 is continuous and of bounded variation, and it is easy to see that it also satisfies the Luzin condition. None of the three function classes on the right-hand side of (3.12) may be omitted. Indeed, the Cantor function 𝜑 and the strict Cantor function 𝜓 may serve as examples of functions in (𝐶 ∩ 𝐵𝑉) \ 𝐴𝐶. Examples of functions in (𝐵𝑉 ∩ 𝐿𝑢) \ 𝐴𝐶 and (𝐶 ∩ 𝐿𝑢) \ 𝐴𝐶 are easily found: Example 3.10. Let 𝑓 : [0, 2] → ℝ be defined by 𝑓 := 𝜒[0,1] . Clearly, 𝑓 satisfies the Luzin condition and, being monotone, also has bounded variation, but is of course discontinuous. On the other hand, let 𝑓 : [0, 1] → ℝ be the function (1.14) from Example 1.8. As we have seen there, 𝑓 is continuous, but not of bounded variation. Now, if 𝑁 ⊂ [0, 1] is a nullset, then ∞

𝑓(𝑁) ⊆ {0} ∪ ⋃ 𝑓(𝑁 ∩ [1/𝑛, 1]) , 𝑛=1

and the set on the right-hand side is a nullset since the function (1.14) is of class 𝐶∞ on [1/𝑛, 1]. So, 𝑓 satisfies the Luzin condition. ♥ We summarize these examples in the following Table 3.1. It is clear that by (3.12), no entries are possible in the diagonal and the last column. Table 3.1. The Vitali–Banach–Zaretskij theorem. Function 𝑓

∈𝐶

∈ 𝐵𝑉

∈ 𝐿𝑢

∈ 𝐴𝐶

∉ 𝐶 ∉ 𝐵𝑉 ∉ 𝐿𝑢 ∉ 𝐴𝐶

– Example 1.8 Example 3.6 Example 3.6

Example 3.10 – Example 3.6 Example 3.6

Example 3.10 Example 1.8 – Example 1.8

– – – –

In Exercise 3.9, we consider a more sophisticated function 𝑓 ∈ 𝐶([0, 1]) ∩ 𝐿𝑢([0, 1]) whose total variation may be made finite or infinite by a suitable choice of free param­ eters. The same reasoning as in Example 3.10 shows that the function 𝑓𝛼,𝛽 defined in (0.86) satisfies the Luzin property for all choices of 𝛼 and 𝛽. Combining this with Ex­ ercises 1.8 and 1.9 allows us to precisely determine those values of 𝛼 and 𝛽 for which the function (0.86) is absolutely continuous: Example 3.11. For 𝛼, 𝛽 ∈ ℝ, let 𝑓𝛼,𝛽 : [0, 1] → ℝ be defined as in (0.86). From the above reasoning, we know that 𝑓𝛼,𝛽 satisfies the Luzin property for all choices of 𝛼

216 | 3 Absolutely continuous functions Table 3.2. Spaces 𝐴𝐶𝑝 (𝐼) and 𝑊𝐵𝑉𝑝 (𝐼) over 𝐼 = [𝑎, 𝑏]. 𝐴𝐶(𝐼) ∩ 𝐵𝑉(𝐼)

= =

𝐴𝐶1 (𝐼) ∩ 𝑊𝐵𝑉1 (𝐼)

⊂ ⊂

𝐴𝐶𝑝 (𝐼) ∩ 𝑊𝐵𝑉𝑝 (𝐼)

⊂ ⊂

𝐴𝐶𝑞 (𝐼) ∩ 𝑊𝐵𝑉𝑞 (𝐼)

⊂ ⊂

𝐶(𝐼) ∩ 𝐵(𝐼)

and 𝛽, and from Exercises 1.8 and 1.9 we know that 𝑓𝛼,𝛽 ∈ 𝐵𝑉([0, 1]) if and only if 𝛽 > 0 and 𝛼 + 𝛽 ≥ 0, or 𝛽 ≤ 0 and 𝛼 + 𝛽 > 0. Moreover, Proposition 0.48 shows that 𝑓𝛼,𝛽 is continuous at zero (and so on the whole interval [0, 1]) if and only if 𝛼 > 0 and 𝛽 is arbitrary, or 𝛼 ≤ 0 and 𝛼 + 𝛽 > 0. Thus, from Theorem 3.9, we conclude that 𝑓𝛼,𝛽 ∈ 𝐴𝐶([0, 1]) if and only if 𝛼 + 𝛽 > 0. ♥ Now, we consider a generalization of absolute continuity due to Love [183] which is usually called 𝑝-absolute continuity. Definition 3.12. Given a collection 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏]) and a function 𝑓 : [𝑎, 𝑏] → ℝ, for 𝑝 ≥ 1, we put, in analogy to (3.1), 𝑛

1/𝑝 𝑝

𝛩𝑝 (𝑆) := { ∑ |𝑏𝑘 − 𝑎𝑘 | } 𝑘=1

𝑛

,

1/𝑝 𝑝

𝛤𝑝 (𝑓; 𝑆) := { ∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| }

.

(3.13)

𝑘=1

We call the function 𝑓 𝑝-absolutely continuous on [𝑎, 𝑏] and write 𝑓 ∈ 𝐴𝐶𝑝 ([𝑎, 𝑏]) if for each 𝜀 > 0, there exists some 𝛿 > 0 such that 𝛤𝑝 (𝑓; 𝑆) ≤ 𝜀 for any 𝑆 ∈ 𝛴([𝑎, 𝑏]) satisfying 𝛩𝑝 (𝑆) ≤ 𝛿. ◼ Of course, for 𝑝 = 1, we get the old Definition 3.1, which means that 𝐴𝐶1 = 𝐴𝐶. In the same way as we have proved the inclusion 𝐴𝐶([𝑎, 𝑏]) ⊆ 𝐵𝑉([𝑎, 𝑏]) in Proposition 1.22, one may show that⁶ 𝐶([𝑎, 𝑏]) ∩ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊆ 𝐴𝐶𝑞 ([𝑎, 𝑏]) ⊆ 𝐶([𝑎, 𝑏]) ∩ 𝑊𝐵𝑉𝑞 ([𝑎, 𝑏])

(3.14)

for 1 ≤ 𝑝 < 𝑞 < ∞, where 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) denotes the Wiener space introduced in Definition 1.31. Moreover, the spaces 𝐴𝐶𝑝 ([𝑎, 𝑏]) are increasing in 𝑝; more precisely, 𝐴𝐶([𝑎, 𝑏]) = 𝐴𝐶1 ([𝑎, 𝑏]) ⊆ 𝐴𝐶𝑝 ([𝑎, 𝑏]) ⊆ 𝐴𝐶𝑞 ([𝑎, 𝑏]) ⊆ 𝐶([𝑎, 𝑏])

(3.15)

for 1 ≤ 𝑝 ≤ 𝑞 < ∞ which is parallel to (1.72). We summarize all of these inclusions in Table 3.2 above. Following [183] (see also [184, 185]), we now consider a class of functions which is similar to the class 𝐶𝐵𝑉 introduced in Section 1.2.

6 Exercise 3.25 shows that the first inclusion in (3.14) is strict in case 𝑝 < 𝑞, while Exercise 3.26 shows that the second inclusion in (3.15) is strict.

3.2 The Vitali–Banach–Zaretskij theorem | 217

Definition 3.13. For 𝑝 > 1, we denote by 𝐶𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) the set of all continuous func­ tions 𝑓 : [𝑎, 𝑏] → ℝ satisfying {𝑚 } inf { ∑ Var𝑊 𝑝 (𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) : {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏])} = 0 , {𝑗=1 }

(3.16)

where Var𝑊 𝑝 (𝑓; [𝛼, 𝛽]) denotes the Wiener 𝑝-variation (1.61) of 𝑓 on [𝛼, 𝛽], and the infi­ mum in (3.16) is taken over all partitions of [𝑎, 𝑏]. ◼ This definition is not made for 𝑝 = 1 as it would admit only constant functions. How­ ever, for 𝑝 > 1, it turns out to be something we already know: Proposition 3.14. The equality 𝐶𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) = 𝐴𝐶𝑝 ([𝑎, 𝑏])

(3.17)

holds for 𝑝 > 1. Proof. Suppose first that 𝑓 ∈ 𝐶𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]), and let 𝜀 > 0. By definition, we may then find a partition {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) satisfying 𝑚 𝜀 𝑝 ∑ Var𝑊 𝑝 (𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) ≤ ( ) . 2 𝑗=1

Let 𝛿 := min {𝑡1 − 𝑡0 , 𝑡2 − 𝑡1 , . . . , 𝑡𝑚 − 𝑡𝑚−1 }. Now, if 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏]) is any collection satisfying 𝛩𝑝 (𝑆) ≤ 𝛿, see (3.13), not more than one 𝑡𝑗 lies in each of the subintervals of 𝑆. Let 𝜏𝑘 be the 𝑡𝑗 in [𝑎𝑘 , 𝑏𝑘 ] if there is one. Otherwise, let 𝜏𝑘 := 𝑎𝑘 . Then we obtain 𝑛

𝛤𝑝 (𝑓; 𝑆)𝑝 = ∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|𝑝 𝑘=1 𝑛

𝑛

≤ 2𝑝−1 ∑ |𝑓(𝑏𝑘 ) − 𝑓(𝜏𝑘 )|𝑝 + 2𝑝−1 ∑ |𝑓(𝜏𝑘 ) − 𝑓(𝑎𝑘 )|𝑝 𝑘=1 𝑚

𝑘=1

𝑝 ≤ 2𝑝 ∑ Var𝑊 𝑝 (𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) ≤ 𝜀 , 𝑗=1

which shows that 𝑓 ∈ 𝐴𝐶𝑝 ([𝑎, 𝑏]). Conversely, suppose now that 𝑓 ∈ 𝐴𝐶𝑝 ([𝑎, 𝑏]) (so 𝑓 ∈ 𝐶([𝑎, 𝑏])), and let 𝜀 > 0. Choose 𝛿 > 0 such that 𝛤𝑝 (𝑓; 𝑆) ≤ 𝜀 for all collections 𝑆 ∈ 𝛴([𝑎, 𝑏]) satisfying 𝛩𝑝 (𝑆) ≤ (𝑏 − 𝑎)1/𝑝 𝛿1−1/𝑝 . Let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) be a fixed partition with 𝜇(𝑃) ≤ 𝛿, where 𝜇(𝑃) denotes the mesh size (1.2) of 𝑃. Taking 𝑎𝑘 := 𝑡𝑘−1 and 𝑏𝑘 := 𝑡𝑘 for 𝑘 = 1, 2, . . . , 𝑚, we then have 𝑚

𝑚

𝑗=1

𝑘=1

∑ (𝑡𝑗 − 𝑡𝑗−1 )𝑝 ≤ ∑ 𝛿𝑝−1 (𝑏𝑘 − 𝑎𝑘 ) ≤ 𝛿𝑝−1 (𝑏 − 𝑎)

218 | 3 Absolutely continuous functions which means that 𝑆 = {[𝑡0 , 𝑡1 ], [𝑡1 , 𝑡2 ] . . . , [𝑡𝑚−1 , 𝑡𝑚 ]} ∈ 𝛴([𝑎, 𝑏]) satisfies 𝛩𝑝 (𝑆)𝑝 ≤ (𝑏 − 𝑎)𝛿𝑝−1 . By assumption, we then have 𝑚

𝑝 𝑝 ∑ Var𝑊 𝑝 (𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) = 𝛤𝑝 (𝑓; 𝑆) ≤ 𝜀

𝑗=1

and so 𝑓 ∈ 𝐶𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]). Another characterization of the class 𝐴𝐶𝑝 for 𝑝 > 1 may be found in Theorem 3.41 in Section 3.7 below.

3.3 Reconstructing a function from its derivative This section will be dedicated to the following two questions: – Given 𝑓 : [𝑎, 𝑏] → ℝ, when may we write 𝑓 as an “indefinite integral,” i.e. in the form 𝑥

𝑓(𝑥) = 𝑐 + ∫ 𝑔(𝑡) 𝑑𝑡 𝑎

–

for some constant 𝑐 ∈ ℝ and some function 𝑔 ∈ 𝐿 1 ([𝑎, 𝑏])? Given 𝑓 : [𝑎, 𝑏] → ℝ with 𝑓󸀠 ∈ 𝐿 1 ([𝑎, 𝑏]), is it true that 𝑓 and the function 𝑥

̃ := ∫ 𝑓󸀠 (𝑡) 𝑑𝑡 𝑓(𝑥) 𝑎

differ only by some constant? It turns out that absolute continuity is precisely what we need here. In particular, the answer to the first question is suggested by Proposition 3.4, while the answer to the second question is closely related to the problem of reconstructing a function from its derivative. To put things in the right framework, let us recall how this may be done very easily for 𝐶1 functions. Given a continuously differentiable function 𝑓 : [𝑎, 𝑏] → ℝ, the fundamental theorem of calculus for the Riemann integral states that 𝑥

∫ 𝑓󸀠 (𝑡) 𝑑𝑡 = 𝑓(𝑥) − 𝑓(𝑎)

(3.18)

𝑎

for each 𝑥 ∈ [𝑎, 𝑏], where the integral in (3.18) is the Riemann integral; in particular, 𝑏

∫ 𝑓󸀠 (𝑡) 𝑑𝑡 = 𝑓(𝑏) − 𝑓(𝑎) . 𝑎

(3.19)

3.3 Reconstructing a function from its derivative |

219

Conversely, given a Riemann integrable (in particular, bounded) function 𝑔 : [𝑎, 𝑏] → ℝ, the function 𝑓 defined by 𝑥

𝑓(𝑥) := ∫ 𝑔(𝑡) 𝑑𝑡

(𝑎 ≤ 𝑥 ≤ 𝑏)

(3.20)

𝑎

is Lipschitz continuous; moreover, in case of a continuous integrand 𝑔 we even have 𝑓 ∈ 𝐶1 ([𝑎, 𝑏]) and 𝑓󸀠 (𝑥) = 𝑔(𝑥) for all 𝑥 ∈ [𝑎, 𝑏]. However, the two questions stated above refer to the Lebesgue integral, and here the corresponding problem is more delicate. First of all, if the integral in (3.18) (or (3.19) is the Lebesgue integral, it suffices that 𝑓󸀠 exists only a.e. on [𝑎, 𝑏]. However, the equality (3.19) may then fail for two different reasons: the derivative 𝑓󸀠 need not be integrable, and even if it is, (3.19) need not hold. We illustrate both phenomena by an example. Example 3.15. Consider the function (0.86) for 𝛼 = 2 and 𝛽 = −2, i.e. {𝑥2 sin 𝑓2,−2 (𝑥) := { 0 {

1 𝑥2

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 .

(3.21)

Then 𝑓2,−2 is everywhere differentiable on [0, 1] with {2𝑥 sin 𝑥12 − 󸀠 𝑓2,−2 (𝑥) := { 0 {

2 𝑥

cos 𝑥12

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 .

(3.22)

The first term in (3.22) is continuous and therefore belongs to 𝐿 1 ([0, 1]). However, the second term is of the form⁷ (0.86) with 𝛼 = −1 and 𝛽 = −2, and Exercise 0.7 shows ♥ that this term does not belong to 𝐿 1 ([0, 1]). The behavior of the function in Example 3.15 is explained by the fact that the choice (𝛼, 𝛽) = (2, −2) is not covered by the condition given in Example 3.11, i.e. 𝑓2,−2 ∈ ̸ 𝐴𝐶([0, 1]). We will come back to this in a moment (see Theorem 3.18 below). First, we consider another example, where the integral in (3.19) exists, but is not equal to 𝑓(𝑏) − 𝑓(𝑎). Example 3.16. Let 𝜑 : [0, 1] → ℝ be the Cantor function from Example 3.6. As we have seen there, 𝜑 ∈ ̸ 𝐴𝐶([0, 1]). Since 𝜑󸀠 (𝑡) = 0 a.e. on [0, 1] (namely, for all 𝑡 ∈ [0, 1] \ 𝐶), we have 𝜑󸀠 ∈ 𝐿 1 ([0, 1]) and 1

∫ 𝜑󸀠 (𝑡) 𝑑𝑡 = 0 . 0

On the other hand, 𝜑(1) − 𝜑(0) = 1, by definition of 𝜑.

♥

7 In (0.86), we have the sine function instead, but the cosine function (0.88) of course has the same behavior near zero.

220 | 3 Absolutely continuous functions The following important theorem shows that a one-sided estimate holds in (3.19) if 𝑓 is monotonically increasing. Theorem 3.17. If 𝑓 : [𝑎, 𝑏] → ℝ is increasing, then the derivative 𝑓 exists a.e. on [𝑎, 𝑏], belongs to 𝐿 1 ([𝑎, 𝑏]), and satisfies 𝑏

∫ 𝑓󸀠 (𝑡) 𝑑𝑡 ≤ 𝑓(𝑏) − 𝑓(𝑎) .

(3.23)

𝑎

Proof. By Exercise 1.29, 𝑓 is a.e. differentiable on [𝑎, 𝑏]. We extend 𝑓 to the larger do­ main [𝑎−1, 𝑏+1] by putting 𝑓(𝑡) = 𝑓(𝑎) for 𝑎−1 ≤ 𝑡 < 𝑎 and 𝑓(𝑡) = 𝑓(𝑏) for 𝑏 < 𝑡 ≤ 𝑏+1, and define functions 𝑓𝑛 , 𝑔𝑛 , and 𝑔 by 𝑓𝑛 (𝑡) := 𝑓(𝑡 + 𝑛1 ), and 𝑔(𝑡) := lim

ℎ→0

𝑔𝑛 (𝑡) := 𝑛 [𝑓𝑛 (𝑡) − 𝑓(𝑡)] ,

(3.24)

𝑓(𝑡 + ℎ) − 𝑓(𝑡) . ℎ

(3.25)

Since 𝑓 is differentiable a.e., 𝑔 is a.e. defined on [𝑎, 𝑏]. Moreover, 𝑔 is measurable since both 𝑓 and 𝑓𝑛 are increasing, and (𝑔𝑛 )𝑛 converges a.e. on [𝑎, 𝑏] to 𝑔. By Fatou’s theorem (Theorem 0.5), we have 𝑏

𝑏

𝑏

∫ 𝑔(𝑡) 𝑑𝑡 ≤ lim inf ∫ 𝑔𝑛 (𝑡) 𝑑𝑡 = lim inf 𝑛 ∫ [𝑓(𝑡 + 𝑛1 ) − 𝑓(𝑡)] 𝑑𝑡 . 𝑛→∞

𝑎

𝑛→∞

𝑎

(3.26)

𝑎

After a simple change of variables for the last integral, we obtain 𝑏+1/𝑛

𝑏

∫ [𝑓(𝑡 +

1 ) 𝑛

𝑏

− 𝑓(𝑡)] 𝑑𝑡 = ∫ 𝑓(𝑡) 𝑑𝑡 − ∫ 𝑓(𝑡) 𝑑𝑡

𝑎

𝑎+1/𝑛

𝑎

𝑏+1/𝑛

𝑎+1/𝑛

𝑎+1/𝑛

= ∫ 𝑓(𝑡) 𝑑𝑡 − ∫ 𝑓(𝑡) 𝑑𝑡 = 𝑎

𝑏

𝑓(𝑏) − ∫ 𝑓(𝑡) 𝑑𝑡 . 𝑛 𝑎

Since 𝑓(𝑡) ≥ 𝑓(𝑎) for 𝑡 ∈ [𝑎, 𝑎 + 1/𝑛], we further get 𝑎+1/𝑛

𝑎+1/𝑛

∫ 𝑓(𝑡) 𝑑𝑡 ≥ ∫ 𝑓(𝑎) 𝑑𝑡 = 𝑎

𝑎

𝑓(𝑎) , 𝑛

and combining this with (3.26) and (3.27) yields 𝑏

∫ 𝑔(𝑡) 𝑑𝑡 ≤ lim inf 𝑛 ( 𝑎

The proof is complete.

𝑛→∞

𝑓(𝑏) 𝑓(𝑎) − ) = 𝑓(𝑏) − 𝑓(𝑎) . 𝑛 𝑛

(3.27)

3.3 Reconstructing a function from its derivative | 221

As we have seen, an example for strict inequality in (3.23) is the monotone Cantor function 𝜑. The next theorem shows that this is not accidental. Theorem 3.18. If 𝑓 : [𝑎, 𝑏] → ℝ is absolutely continuous, then the derivative 𝑓󸀠 exists a.e. on [𝑎, 𝑏], belongs to 𝐿 1 ([𝑎, 𝑏]), and satisfies the equality (3.19). Proof. Since every absolutely continuous function has bounded variation, the deriva­ tive 𝑓󸀠 of 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) exists a.e. in [𝑎, 𝑏]. Moreover, by Proposition 3.3, we may as­ sume without loss of generality that 𝑓 is increasing. We define 𝑔𝑛 : [𝑎, 𝑏] → ℝ as in (3.24); then, 𝑔𝑛 is continuous, and again 𝑔𝑛 (𝑡) → 𝑓󸀠 (𝑡) a.e. on [𝑎, 𝑏]. As in (3.27), we get 𝑎+1/𝑛

𝑏

∫ 𝑔𝑛 (𝑡) 𝑑𝑡 = 𝑓(𝑏) − 𝑛 ∫ 𝑓(𝑡) 𝑑𝑡 , 𝑎

𝑎

where the integral on the right-hand side is now the Riemann integral. The continuity of 𝑓 implies that 𝑏

∫ 𝑔𝑛 (𝑡) 𝑑𝑡 → 𝑓(𝑏) − 𝑓(𝑎) (𝑛 → ∞) , 𝑎

by the mean value theorem for Riemann integrals. So, for proving the theorem, it suf­ fices to show that also 𝑏

𝑏

∫ 𝑔𝑛 (𝑡) 𝑑𝑡 → ∫ 𝑓󸀠 (𝑡) 𝑑𝑡 𝑎

(3.28)

(𝑛 → ∞) .

𝑎

Let 𝜀 > 0. Since 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]), we find a 𝛿 > 0 such that 𝛤(𝑓; 𝑆) ≤ 𝜀 for any 𝑆 ∈ 𝛴([𝑎, 𝑏]) with 𝛩(𝑆) ≤ 𝛿. Moreover, by Theorem 0.6, we can choose 𝛿󸀠 ∈ (0, 𝛿) with the property that ∫ 𝑓󸀠 (𝑡) 𝑑𝑡 ≤ 𝜀

(3.29)

𝑀

for any set 𝑀 ⊂ [𝑎, 𝑏] satisfying 𝜆(𝑀) ≤ 𝛿󸀠 . Denote by 𝐷 the set of all points in [𝑎, 𝑏] where 𝑓 is differentiable, so [𝑎, 𝑏] \ 𝐷 is a nullset. By Egorov’s theorem (Theorem 0.1), we find a measurable set 𝑀 ⊆ 𝐷 such that 𝜆(𝑀) ≤ 𝛿󸀠 and (𝑔𝑛 )𝑛 converges uniformly to 𝑓󸀠 on 𝐷 \ 𝑀. This implies, in particular, that there exists 𝑛0 ∈ ℕ such that ∫ |𝑔𝑛 (𝑡) − 𝑓󸀠 (𝑡)| 𝑑𝑡 ≤ 𝜀

(3.30)

𝐷\𝑀

for 𝑛 ≥ 𝑛0 . So, for these 𝑛, we have, by (3.29) and (3.30), 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑏 𝑏 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󸀠 󸀠 󸀠 󵄨󵄨󵄨∫ 𝑔𝑛 (𝑡) 𝑑𝑡 − ∫ 𝑓 (𝑡) 𝑑𝑡󵄨󵄨󵄨 = 󵄨󵄨󵄨∫ 𝑔𝑛 (𝑡) 𝑑𝑡 − ∫ 𝑓 (𝑡) 𝑑𝑡󵄨󵄨󵄨 ≤ ∫ |𝑔𝑛 (𝑡) − 𝑓 (𝑡)| 𝑑𝑡 󵄨 󵄨 󵄨 󵄨󵄨 𝑎 󵄨 󵄨 󵄨 𝑎 𝐷 󵄨 󵄨𝐷 󵄨 𝐷 󵄨 = ∫ |𝑔𝑛 (𝑡) − 𝑓󸀠 (𝑡)| 𝑑𝑡 + ∫ |𝑔𝑛 (𝑡) − 𝑓󸀠 (𝑡)| 𝑑𝑡 ≤ 2𝜀 + ∫ |𝑔𝑛 (𝑡)| 𝑑𝑡 . 𝐷\𝑀

𝑀

𝑀

222 | 3 Absolutely continuous functions Thus, to prove (3.28), it remains to show that the last integral tends to zero as 𝑛 → ∞. Since 𝜆(𝑀) ≤ 𝛿󸀠 < 𝛿, there exists an open set 𝑂 ⊂ (𝑎, 𝑏) such that 𝑀 ⊆ 𝑂 and 𝜆(𝑂) < 𝛿. We again use the fact that we may write 𝑂 as a countable union of disjoint open intervals ∞

𝑂 = ⋃(𝑎𝑘 , 𝑏𝑘 ) . 𝑘=1

Now, for any 𝜏 ∈ [0, 1] and any 𝑚 ∈ ℕ, the collection of intervals 𝑆 := {[𝑎1 + 𝜏, 𝑏1 + 𝜏], [𝑎2 + 𝜏, 𝑏2 + 𝜏], . . . , [𝑎𝑚 + 𝜏, 𝑏𝑚 + 𝜏]} ∈ 𝛴([𝑎, 𝑏]) satisfies

𝑚

𝛩(𝑆) = ∑ |𝑏𝑘 − 𝑎𝑘 | ≤ 𝛿 . 𝑘=1

Consequently,

𝑚

𝛤(𝑓; 𝑆) = ∑ [𝑓(𝑏𝑘 + 𝜏) − 𝑓(𝑎𝑘 + 𝜏)] ≤ 𝜀 , 𝑘=1

by our choice of 𝛿. However, for 𝑘 = 1, 2, . . . , 𝑚, we have 𝑏𝑘

𝑏𝑘 +1/𝑛

𝑎𝑘 +1/𝑛

∫ 𝑔𝑛 (𝑡) 𝑑𝑡 = 𝑛 ( ∫ 𝑓(𝑡) 𝑑𝑡 − ∫ 𝑓(𝑡) 𝑑𝑡) 𝑎𝑘

𝑎𝑘

𝑏𝑘 1/𝑛

= 𝑛 ∫ [𝑓(𝑏𝑘 + 𝜏) − 𝑓(𝑎𝑘 + 𝜏)] 𝑑𝜏 . 0

Consequently, 𝑚

𝑏𝑘

∫ 𝑔𝑛 (𝑡) 𝑑𝑡 ≤ ∫ 𝑔𝑛 (𝑡) 𝑑𝑡 = ∑ ∫ 𝑔𝑛 (𝑡) 𝑑𝑡 𝑀

𝑘=1 𝑎𝑘

𝑂 1/𝑛

𝑚

= 𝑛 ∫ ( ∑ [𝑓(𝑏𝑘 + 𝜏) − 𝑓(𝑎𝑘 + 𝜏)]) 𝑑𝜏 ≤ 𝜀 , 0

𝑘=1

from which (3.28) follows. We may summarize Proposition 3.4 and Theorem 3.18 in the following way: a function 𝑓 : [𝑎, 𝑏] → ℝ is absolutely continuous if and only if there exists 𝑔 ∈ 𝐿 1 ([𝑎, 𝑏]) such that 𝑥

𝑓(𝑥) = 𝑓(𝑎) + ∫ 𝑔(𝑡) 𝑑𝑡 .

(3.31)

𝑎

Moreover, 𝑓󸀠 (𝑥) = 𝑔(𝑥) a.e. on [𝑎, 𝑏] in this case. This is the fundamental theorem of calculus for the Lebesgue integral.

3.3 Reconstructing a function from its derivative

| 223

We have even more to say on the subject. An absolutely continuous function 𝑓 cannot only be expressed through the integral (3.18) of its derivative 𝑓󸀠 , but also its total variation (which is finite by the inclusion 𝐴𝐶 ⊆ 𝐵𝑉) may be expressed by the integral of |𝑓󸀠 |. This is the contents of the following Theorem 3.19. Let 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]). Then 𝑏

Var(𝑓; [𝑎, 𝑏]) ≥ ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 ,

(3.32)

𝑎

where Var(𝑓; [𝑎, 𝑏]) denotes the total variation (1.4) of 𝑓 on [𝑎, 𝑏]. In case 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]), we even have equality 𝑏

Var(𝑓; [𝑎, 𝑏]) = ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 .

(3.33)

𝑎

Proof. Given 𝜀 > 0, we choose 𝛿 > 0 such that ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 ≤ 𝜀 𝑀

for any set 𝑀 ⊂ [𝑎, 𝑏] satisfying 𝜆(𝑀) ≤ 𝛿, which is possible by Theorem 0.6. Denoting 𝐷+ := {𝑡 : 𝑎 < 𝑡 < 𝑏, 𝑓󸀠 (𝑡) ≥ 0} and 𝐷− := {𝑡 : 𝑎 < 𝑡 < 𝑏, 𝑓󸀠 (𝑡) < 0}, we may find an open set 𝑂 ⊂ (𝑎, 𝑏) satisfying⁸ 𝑚

𝑂 = ⋃(𝑎𝑗 , 𝑏𝑗 ), 𝑗=1

𝜆(𝑂△𝐷+ ) ≤ 𝛿 .

Here, we may assume without loss of generality that 𝑎1 < 𝑏1 < 𝑎2 < . . . < 𝑏𝑚−1 < 𝑎𝑚 < 𝑏𝑚 . Adding, if necessary, the points 𝑏0 := 𝑎 and 𝑎𝑚+1 := 𝑏, we thus obtain a partition 𝑃 := {𝑏0 , 𝑎1 , 𝑏1 , 𝑎2 , . . . , 𝑏𝑚−1 , 𝑎𝑚 , 𝑏𝑚 , 𝑎𝑚+1 } ∈ P([𝑎, 𝑏]). Calculating the variation (1.3) of 𝑓 with respect to this partition and observing that 𝐷− △((𝑎, 𝑏) \ 𝑂) = (𝐷∩ 𝑂) ∪ [(𝑎, 𝑏) \ 𝑂] \ 𝐷− = 𝑂△𝐷+ ,

8 As usual, 𝐴△𝐵 = (𝐴 ∪ 𝐵) \ (𝐴 ∩ 𝐵) = (𝐴 \ 𝐵) ∪ (𝐵 \ 𝐴) denotes the symmetric difference of 𝐴 and 𝐵.

224 | 3 Absolutely continuous functions we get 󵄨󵄨 𝑏𝑗 󵄨󵄨 󵄨󵄨󵄨 𝑚+1 󵄨󵄨󵄨 𝑎𝑘 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󸀠 󸀠 󵄨 Var(𝑓, 𝑃; [𝑎, 𝑏]) = ∑ 󵄨󵄨󵄨∫ 𝑓 (𝑡) 𝑑𝑡󵄨󵄨󵄨 + ∑ 󵄨󵄨󵄨 ∫ 𝑓 (𝑡) 𝑑𝑡󵄨󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨 𝑗=1 󵄨𝑎 󵄨󵄨 𝑘=1 󵄨󵄨󵄨𝑏𝑘−1 󵄨󵄨 󵄨󵄨 𝑗 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨 ≥ 󵄨󵄨󵄨∫ 𝑓󸀠 (𝑡) 𝑑𝑡󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨 ∫ 𝑓󸀠 (𝑡) 𝑑𝑡󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨(𝑎,𝑏)\𝑂 󵄨𝑂 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󸀠 󸀠 󸀠 |𝑓󸀠 (𝑡)| 𝑑𝑡 ≥ 󵄨󵄨󵄨 ∫ 𝑓 (𝑡) 𝑑𝑡󵄨󵄨󵄨 − ∫ |𝑓 (𝑡)| 𝑑𝑡 + 󵄨󵄨󵄨 ∫ 𝑓 (𝑡) 𝑑𝑡󵄨󵄨󵄨󵄨 − ∫ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝐷+ △𝑂 󵄨󵄨𝐷− 󵄨󵄨 𝐷− △((𝑎,𝑏)\𝑂) 󵄨󵄨𝐷+ 𝑚

𝑏

𝑏 󸀠

󸀠

≥ ∫ |𝑓 (𝑡)| 𝑑𝑡 − ∫ |𝑓 (𝑡)| 𝑑𝑡 ≥ ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 − 𝜀 . 𝑎

𝑂△𝐷+

𝑎

Since 𝜀 > 0 was arbitrary, we conclude that 𝑏

Var(𝑓; [𝑎, 𝑏]) ≥ Var(𝑓, 𝑃; [𝑎, 𝑏]) ≥ ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 .

(3.34)

𝑎

The converse inequality follows for 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) from (3.18). In fact, for arbitrary partitions 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), we obtain 󵄨󵄨 󵄨󵄨 𝑡𝑗 𝑏 󵄨󵄨 𝑚 𝑡𝑗 󵄨󵄨 󵄨󵄨 󵄨󵄨 󸀠 Var(𝑓, 𝑃; [𝑎, 𝑏]) = ∑ 󵄨󵄨󵄨 ∫ 𝑓 (𝑡) 𝑑𝑡󵄨󵄨󵄨 ≤ ∑ ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 = ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 , 󵄨󵄨 𝑗=1 󵄨 𝑗=1 󵄨󵄨𝑡 󵄨󵄨 𝑎 𝑡𝑗−1 󵄨󵄨 𝑗−1 󵄨 𝑚

and hence Var(𝑓; [𝑎, 𝑏]) ≤ ‖𝑓󸀠 ‖𝐿 1 , which, together with (3.34), proves the statement. Theorem 3.19 admits a certain refinement which states that we have equality in (3.32) precisely for 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]), see Exercise 3.33. Obviously, in the same way as we proved formula (3.32), we may show that 𝑥

𝑉𝑓 (𝑥) ≥ ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡

(𝑎 ≤ 𝑥 ≤ 𝑏)

(3.35)

𝑎

if 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), where 𝑉𝑓 denotes the variation function (1.13). As before, in case 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]), we even have equality 𝑥

𝑉𝑓 (𝑥) = ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡

(𝑎 ≤ 𝑥 ≤ 𝑏) .

(3.36)

𝑎

As a consequence of Theorem 3.18, we may also obtain the following characteri­ zation of Lipschitz continuous functions.

3.3 Reconstructing a function from its derivative

| 225

Theorem 3.20. A function 𝑓 belongs to 𝐿𝑖𝑝([𝑎, 𝑏]) if and only if 𝑓 may be written in the form 𝑥

(3.37)

𝑓(𝑥) = 𝑐 + ∫ 𝑔(𝑡) 𝑑𝑡 , 𝑎

where 𝑐 ∈ ℝ and 𝑔 ∈ 𝐿 ∞ ([𝑎, 𝑏]). Proof. For 𝑓 ∈ 𝐿𝑖𝑝([𝑎, 𝑏]) ⊆ 𝐴𝐶([𝑎, 𝑏]), we conclude from Theorem 3.18 that (3.19) holds true, so we have (3.37) with 𝑐 := 𝑓(𝑎) and 𝑔 := 𝑓󸀠 . Moreover, from |𝑓(𝑥) − 𝑓(𝑦)| ≤ 𝐿|𝑥 − 𝑦|, it follows that |𝑓󸀠 (𝑡)| ≤ 𝐿 at any point 𝑡 ∈ [𝑎, 𝑏] where 𝑓 is differentiable, and so 𝑓󸀠 ∈ 𝐿 ∞ ([𝑎, 𝑏]) with ‖𝑓󸀠 ‖𝐿 ∞ ≤ 𝑙𝑖𝑝(𝑓; [𝑎, 𝑏]). Conversely, suppose that 𝑓 satisfies (3.37) for some 𝑐 ∈ ℝ and 𝑔 ∈ 𝐿 ∞ ([𝑎, 𝑏]). Then for 𝑎 ≤ 𝑥 < 𝑦 ≤ 𝑏, we obtain 󵄨󵄨 𝑦 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 |𝑓(𝑥) − 𝑓(𝑦)| ≤ 󵄨󵄨∫ 𝑔(𝑡) 𝑑𝑡󵄨󵄨󵄨 ≤ ‖𝑔‖𝐿 ∞ |𝑥 − 𝑦| , 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑥 󵄨󵄨 and so 𝑓 ∈ 𝐿𝑖𝑝([𝑎, 𝑏]) with 𝑙𝑖𝑝(𝑓; [𝑎, 𝑏]) ≤ ‖𝑔‖𝐿 ∞ as claimed. Finally, we mention another consequence of Theorem 3.18 which we will need in what follows and which uses the concept of singular function in the sense of Definition 3.7. Proposition 3.21. Let 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]). Then 𝑓 may be represented as sum 𝑓(𝑥) = 𝑓ac (𝑥) + 𝑓sg (𝑥)

(𝑎 ≤ 𝑥 ≤ 𝑏) ,

(3.38)

where 𝑓ac is absolutely continuous and 𝑓sg is singular or 𝑓sg (𝑥) ≡ 0. Moreover, these functions are uniquely determined within additive constants, and so the representation (3.38) may be made unique by requiring that 𝑓(𝑎) = 𝑓ac (𝑎). Proof. Being of bounded variation, the function 𝑓 admits a derivative a.e. on [𝑎, 𝑏], see Exercise 1.29. From Proposition 3.4, we know that the function 𝑥

𝑓ac (𝑥) = 𝑓(𝑎) + ∫ 𝑓󸀠 (𝑡) 𝑑𝑡

(3.39)

𝑎

is absolutely continuous; moreover, 𝑓ac (𝑎) = 𝑓(𝑎). The function 𝑓sg : [𝑎, 𝑏] → ℝ de­ 󸀠 fined by 𝑓sg (𝑥) := 𝑓(𝑥) − 𝑓ac (𝑥) belongs to 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]) and satisfies 𝑓sg (𝑥) := 󸀠 󸀠 𝑓 (𝑥) − 𝑓ac (𝑥) = 0 a.e. on [𝑎, 𝑏]. So, 𝑓sg is a singular function in the sense of Defini­ tion 3.7, unless 𝑓sg (𝑥) ≡ 0. To prove uniqueness, suppose that 𝑓(𝑥) = 𝑓ac (𝑥) + 𝑓sg (𝑥) = 𝑓aĉ (𝑥) + 𝑓sĝ (𝑥)

(𝑎 ≤ 𝑥 ≤ 𝑏) ,

(3.40)

where both 𝑓ac and 𝑓aĉ are absolutely continuous with 𝑓ac (𝑎) = 𝑓aĉ (𝑎) = 𝑓(𝑎), and 󸀠 󸀠̂ − 𝑓ac )(𝑥) ≡ 0, and so 𝑓ac − 𝑓aĉ is constant, see both 𝑓sg and 𝑓sĝ are singular. Then (𝑓ac ̂ (𝑥), and so Proposition 3.33 below. However, 𝑓ac (𝑎) = 𝑓aĉ (𝑎) implies that 𝑓ac (𝑥) ≡ 𝑓ac ̂ 𝑓sg (𝑥) ≡ 𝑓sg (𝑥) as well.

226 | 3 Absolutely continuous functions In Theorem 1.26, we have established some relations between the properties of the variation function 𝑉𝑓 given in (1.13) and its parent function 𝑓. Now, we come back to this problem with respect to differentiability. As we have seen in Exercises 1.50 and 1.51, it is not true that in analogy to Proposition 1.7, the differentiability of 𝑓 at some point 𝑥0 ∈ (𝑎, 𝑏) implies the differentiability of 𝑉𝑓 at 𝑥0 , or vice versa. However, the following result was proved in [149] and generalized in [281]. Proposition 3.22. Suppose that 𝑓 is differentiable on [𝑎, 𝑏] with bounded derivative. Then 𝑉𝑓 is differentiable a.e. on [𝑎, 𝑏], and the equality 𝑉𝑓󸀠 (𝑥) = |𝑓󸀠 (𝑥)|

(3.41)

holds at all points where 𝑉𝑓󸀠 exists. Proof. Having a bounded derivative on [𝑎, 𝑏], by Theorem 3.20, we have 𝑓∈ 𝐿𝑖𝑝([𝑎, 𝑏]) ⊆ 𝐴𝐶([𝑎, 𝑏]). Moreover, (3.33) implies that 𝑥

𝑉𝑓 (𝑥) = ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 𝑎

for each 𝑥 ∈ [𝑎, 𝑏], and so (3.41) holds a.e. in [𝑎, 𝑏]. If 𝑓 is of bounded variation, then both 𝑓 and 𝑉𝑓 are differentiable a.e. on [𝑎, 𝑏]. How­ ever, the following example shows that the set of points 𝑥 ∈ [𝑎, 𝑏] at which 𝑓󸀠 (𝑥) exists is not necessarily the same as the set of points 𝑥 ∈ [𝑎, 𝑏] at which 𝑉𝑓󸀠 (𝑥) exists; compare this with Exercise 1.51. Example 3.23. Consider the function (0.86) for 𝛼 = 2 and 𝛽 = −1, i.e. {𝑥2 sin 𝑥1 𝑓2,−1 (𝑥) := { 0 {

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 .

Then 𝑓2,−1 is everywhere differentiable on [0, 1] with {2𝑥 sin 𝑥1 − cos 𝑥1 󸀠 𝑓2,−1 (𝑥) := { 0 {

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 .

󸀠 Since the derivative 𝑓2,−1 is bounded, but discontinuous on [0, 1], we have 𝑓2,−1 ∈ 𝐿𝑖𝑝([0, 1]) ⊆ 𝐴𝐶([0, 1]), by Theorem 3.20, but 𝑓2,−1 ∈ ̸ 𝐶1 ([0, 1]). From (3.36), it follows that 𝑥 𝑥 󵄨󵄨 1 󵄨󵄨 1 󸀠 𝑉𝑓 (𝑥) = ∫ |𝑓 (𝑡)| 𝑑𝑡 = ∫ 󵄨󵄨󵄨󵄨2𝑡 sin − cos 󵄨󵄨󵄨󵄨 𝑑𝑡 . 𝑡 𝑡󵄨 󵄨 0

0

However, considering suitable sequences (𝑥𝑛 )𝑛 of “maximal oscillation,” one may show that 𝑉𝑓 is not differentiable at zero. ♥

3.3 Reconstructing a function from its derivative | 227

As a further application of our results, we consider now two norms on 𝐴𝐶 which both make 𝐴𝐶 complete. Proposition 3.24. The linear space 𝐴𝐶([𝑎, 𝑏]), equipped with either the norm ‖𝑓‖𝐴𝐶 := ‖𝑓‖𝐵𝑉

(3.42)

⦀𝑓⦀𝐴𝐶 := ‖𝑓‖𝐿 1 + ‖𝑓󸀠 ‖𝐿 1

(3.43)

or the norm is a Banach space. Proof. To prove the first statement, we have to show that 𝐴𝐶([𝑎, 𝑏]) is closed in 𝐵𝑉([𝑎, 𝑏]) with respect to the norm (1.16). So, let (𝑓𝑛 )𝑛 be a sequence in 𝐴𝐶([𝑎, 𝑏]) satisfying ‖𝑓𝑛 − 𝑓‖𝐵𝑉 → 0, as 𝑛 → ∞, for some 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), where without loss of generality, 𝑓𝑛 (𝑎) = 𝑓(𝑎) = 0. Writing 𝑓 in the form (3.38), with 𝑓ac being absolutely continuous and 𝑓sg being singular (or zero), we get Var(𝑓sg ; [𝑎, 𝑏]) + Var(𝑓ac − 𝑓𝑛 ; [𝑎, 𝑏]) = Var(𝑓 − 𝑓𝑛 ; [𝑎, 𝑏]) → 0 (𝑛 → ∞) . This implies that Var(𝑓sg ; [𝑎, 𝑏]) = 0, and hence 𝑓 = 𝑓ac ∈ 𝐴𝐶([𝑎, 𝑏]) as claimed. To prove the second statement, let (𝑓𝑛 )𝑛 be a Cauchy sequence in 𝐴𝐶([𝑎, 𝑏]) with respect to the norm (3.43); then, (𝑓𝑛 )𝑛 converges in 𝐿 1 ([𝑎, 𝑏]) to some function 𝑓 ∈ 𝐿 1 ([𝑎, 𝑏]), and (𝑓𝑛󸀠 )𝑛 is Cauchy with respect to the 𝐿 1 -norm. We show that 𝑓 is equiva­ lent to an absolutely continuous function and 𝑓𝑛󸀠 → 𝑓󸀠 , as 𝑛 → ∞, in 𝐿 1 ([𝑎, 𝑏]). Since 𝐿 1 ([𝑎, 𝑏]) is complete with respect to the natural norm (0.11), we find some 𝑔 ∈ 𝐿 1 ([𝑎, 𝑏]) such that 𝑓𝑛󸀠 → 𝑔 in the 𝐿 1 -norm. Moreover, since each 𝑓𝑛 is absolutely continuous, we have 𝑥

𝑓𝑛 (𝑥) = 𝑓𝑛 (𝑎) + ∫ 𝑓𝑛󸀠 (𝑡) 𝑑𝑡 , 𝑎

by Theorem 3.18. Define ℎ : [𝑎, 𝑏] → ℝ by 𝑥

ℎ(𝑥) := ∫ 𝑔(𝑡) 𝑑𝑡 . 𝑎

Then ‖𝑓𝑛󸀠 − 𝑔‖𝐿 1 → 0 implies that ℎ(𝑥) = lim (𝑓𝑛 (𝑥) − 𝑓𝑛 (𝑎)) 𝑛→∞

for every 𝑥 ∈ [𝑎, 𝑏]. Finally, from ‖𝑓𝑛 − 𝑓‖𝐿 1 → 0, we conclude that there exists a subsequence of (𝑓𝑛 )𝑛 which converges pointwise a.e. to 𝑓. Consequently, 𝑓 = ℎ + 𝑐 for some constant 𝑐, and 𝑓󸀠 = 𝑔 a.e. on [𝑎, 𝑏]. Therefore, ‖𝑓𝑛󸀠 − ℎ󸀠 ‖𝐿 1 → 0, as 𝑛 → ∞, and we are done.

228 | 3 Absolutely continuous functions Observe that, by (3.33), the norm (3.42) may be written in the form ‖𝑓‖𝐴𝐶 = |𝑓(𝑎)| + ‖𝑓󸀠 ‖𝐿 1 . We close this section with a geometric characterization of bounded variation and absolute continuity of a function 𝑔 : [𝑎, 𝑏] → ℝ taken from the paper [127]. Without loss of generality, we take [𝑎, 𝑏] = [0, 1]. This characterization is related to the integral mean 𝑔ℎ of a function 𝑔 ∈ 𝐿 1 ([0, 1]) which is defined for 0 < ℎ < 1 by⁹ ℎ

1 { ℎ ∫ 𝑔(𝑥 + 𝑡) 𝑑𝑡 𝑔ℎ (𝑥) := { 0ℎ 1 { ℎ ∫0 𝑔(1 − ℎ + 𝑡) 𝑑𝑡

for 0 ≤ 𝑥 ≤ 1 − ℎ ,

(3.44)

for 1 − ℎ < 𝑥 ≤ 1 .

Thus, 𝑔ℎ is defined to be constant on the interval [1 − ℎ, 1]. By a simple change of variables, we may rewrite (3.44) equivalently in the form 𝑥+ℎ

𝑓(𝑥+ℎ)−𝑓(𝑥)

1 { ℎ ∫ 𝑔(𝑠) 𝑑𝑠 = ℎ 𝑔ℎ (𝑥) = { 𝑥1 𝑓(1)−𝑓(1−ℎ) 1 ℎ { ℎ ∫1−ℎ 𝑔(𝑠) 𝑑𝑠 ≡

for 0 ≤ 𝑥 ≤ 1 − ℎ ,

(3.45)

for 1 − ℎ < 𝑥 ≤ 1 ,

where 𝑓 is the “primitive a.e.” of 𝑔 given in (3.20). Theorem 3.25. For 𝑔 ∈ 𝐿 1 ([0, 1]), the function (3.44) has the following properties. (a) For each ℎ ∈ (0, 1), the function 𝑔ℎ is absolutely continuous. (b) We have lim sup ⦀𝑔ℎ ⦀𝐴𝐶 < ∞ , (3.46) ℎ→0+

where ⦀ ⋅ ⦀𝐴𝐶 denotes the norm (3.43), if and only if 𝑔 is equivalent to a function of bounded variation; moreover, Var(𝑔ℎ ; [0, 1]) ≤ Var(𝑔; [0, 1])

(3.47)

lim ⦀𝑔ℎ − 𝑔⦀𝐴𝐶 = 0 ,

(3.48)

in this case. (c) We have ℎ→0+

if and only if 𝑔 is equivalent to an absolutely continuous function. Proof. The assertion (a) follows from (3.45) and Proposition 3.4. To prove (b), suppose that 𝑔 ∈ 𝐵𝑉([0, 1]), and fix ℎ ∈ (0, 1). By the fundamental theorem of calculus and (3.45), we have 𝑔(𝑥 + ℎ) − 𝑔(𝑥) (0 ≤ 𝑥 ≤ 1 − ℎ) (3.49) 𝑔ℎ󸀠 (𝑥) = ℎ

9 In a similar form, such integral means occur in the definition of the integral modulus of continuity and the formulation of compactness criteria, see (0.98) and Proposition 0.55.

3.3 Reconstructing a function from its derivative |

229

a.e. on [0, 1−ℎ], and 𝑔ℎ󸀠 (𝑥) ≡ 0 on [1−ℎ, 1]. Choose 𝑚 ∈ ℕ such that (𝑚−1)ℎ ≤ 1−ℎ < 𝑚ℎ. Then from Theorem 3.19, we deduce that 1−ℎ

Var(𝑔ℎ ; [0, 1]) = ∫

1−ℎ

|𝑔ℎ󸀠 (𝑡)| 𝑑𝑡

0

1 = ∫ |𝑔(𝑡 + ℎ) − 𝑔(𝑡)| 𝑑𝑡 ℎ 0

(3.50)

ℎ

1 𝑚 = ∫ ∑ |𝑔(𝑡 + 𝑗ℎ) − 𝑔(𝑡 + (𝑗 − 1)ℎ)| 𝑑𝑡 . ℎ 𝑗=1 0

For fixed 𝑡, consider the partition 𝑃𝑡 := {𝑡, 𝑡 + ℎ, . . . , 𝑡 + 𝑚ℎ} ∈ P([𝑡, 𝑡 + 𝑚ℎ]). We may then estimate the last integrand in (3.50) by 𝑚

∑ |𝑔(𝑡 + 𝑗ℎ) − 𝑔(𝑡 + (𝑗 − 1)ℎ)| = Var(𝑔, 𝑃𝑡 ; [𝑡, 𝑡 + 𝑚ℎ]) ≤ Var(𝑔; [0, 1]) ,

(3.51)

𝑗=1

and so (3.47) follows. Moreover, given 𝜀 > 0, we may find a 𝛿 > 0 such that 𝑚

∑ |𝑔(𝑡 + 𝑗ℎ) − 𝑔(𝑡 + (𝑗 − 1)ℎ)| > Var(𝑔; [0, 1]) − 𝜀 𝑗=1

provided that 0 < ℎ < 𝛿. Combining this with (3.51), we conclude that Var(𝑔; [0, 1]) − 𝜀 < Var(𝑔ℎ ; [0, 1]) ≤ Var(𝑔; [0, 1]), showing that the upper limit in (3.46) exists and is finite. Conversely, suppose that 𝑔 is not equivalent to any function of bounded variation. Then for each 𝜔 > 0, there is a 𝛿 > 0 such that Var(𝑔ℎ ; [0, 1]) > 𝜔 for all ℎ ∈ (0, 𝛿), which implies that the upper limit (3.46) cannot be finite. Now, we prove assertion (c). Suppose first that 𝑔 ∈ 𝐴𝐶([0, 1]). Since ‖𝑔 − 𝑔ℎ ‖𝐿 1 → 0 as ℎ → 0+, for every function 𝑔 ∈ 𝐿 1 ([0, 1]) (Proposition 0.53), we only have to show that also (3.52) lim ‖𝑔󸀠 − 𝑔ℎ󸀠 ‖𝐿 1 = 0 . ℎ→0+

Since 𝑔 is absolutely continuous, by (3.33) and (3.49), we have 1

1−ℎ

1 Var(𝑔ℎ ; [0, 1]) = ∫ |𝑔(𝑡 + ℎ) − 𝑔(𝑡)| 𝑑𝑡 . ℎ

󸀠

Var(𝑔; [0, 1]) = ∫ |𝑔 (𝑡)| 𝑑𝑡, 0

0

Let 𝜀 > 0. As in the proof of Theorem 3.18, we find a 𝛿 ∈ (0, 1) such that ∫ |𝑔󸀠 (𝑥)| 𝑑𝑥 ≤ 𝜀 𝑀

for any set 𝑀 ⊂ [0, 1] satisfying 𝜆(𝑀) ≤ 𝛿. Consider any sequence (ℎ𝑛 )𝑛 of positive real numbers converging to zero as 𝑛 → ∞. The equality (3.49) then shows that (𝑔ℎ󸀠 𝑛 (𝑥))𝑛

230 | 3 Absolutely continuous functions converges to 𝑔󸀠 (𝑥) at any point 𝑥 of differentiability of 𝑔, i.e. a.e. in [0, 1]. By Egorov’s theorem (Theorem 0.1), we find a set 𝑇 ⊆ [0, 1] of measure 𝜆(𝑇) > 1 − 𝛿 such that (𝑔ℎ󸀠 𝑛 )𝑛 converges even uniformly on 𝑇 to 𝑔󸀠 as 𝑛 → ∞. Choose 𝜂 > 0 such that 0 < ℎ𝑛 < 𝜂 implies ∫ |𝑔ℎ󸀠 𝑛 (𝑥) − 𝑔󸀠 (𝑥)| 𝑑𝑥 ≤ 𝜀 . 𝑇

Now, the estimate ∫ |𝑔ℎ󸀠 𝑛 (𝑥)| 𝑑𝑥 + ∫ |𝑔ℎ󸀠 𝑛 (𝑥)| 𝑑𝑥 ≤ 𝑇

[0,1]\𝑇

∫ |𝑔󸀠 (𝑥)| 𝑑𝑥 + ∫ |𝑔󸀠 (𝑥)| 𝑑𝑥 𝑇

[0,1]\𝑇

shows that ∫ |𝑔ℎ󸀠 𝑛 (𝑥)| 𝑑𝑥 ≤ 𝜀 + ∫ |𝑔󸀠 (𝑥)| 𝑑𝑥 − ∫ |𝑔ℎ󸀠 𝑛 (𝑥)| 𝑑𝑥 𝑇

[0,1]\𝑇

𝑇 󸀠

≤ 𝜀 ∫ |𝑔 (𝑥) −

𝑔ℎ󸀠 𝑛 (𝑥)| 𝑑𝑥

≤ 2𝜀 .

𝑇

It follows that 1

∫ |𝑔󸀠 (𝑥) − 𝑔ℎ󸀠 𝑛 (𝑥)| 𝑑𝑥 ≤ ∫ |𝑔󸀠 (𝑥) − 𝑔ℎ󸀠 𝑛 (𝑥)| 𝑑𝑥 0

𝑇

+ ∫ |𝑔󸀠 (𝑥)| 𝑑𝑥 + ∫ |𝑔ℎ󸀠 𝑛 (𝑥)| 𝑑𝑥 ≤ 4𝜀 . [0,1]\𝑇

[0,1]\𝑇

We conclude that (3.52) is true, and so ⦀𝑔 − 𝑔ℎ ⦀𝐴𝐶 → 0 as ℎ → 0+. The converse argument is proved as in part (b). We illustrate Theorem 3.25 by means of a simple example involving a function 𝑔 ∈ 𝐵𝑉([0, 1]) \ 𝐴𝐶([0, 1]). Example 3.26. Let 𝑔 = 𝜒[1/2,1] : [0, 1] → ℝ be the characteristic function of the in­ terval [1/2, 1]. Clearly, this function has bounded variation, being monotonically in­ creasing, but is not equivalent to an absolutely continuous function. By Theorem 3.25, this means that lim sup ⦀𝑔ℎ ⦀𝐴𝐶 < ∞, ℎ→0+

⦀𝑔 − 𝑔ℎ ⦀𝐴𝐶 ↛ 0 (ℎ → 0+) .

(3.53)

Since only small values of ℎ are important for proving (3.53), we take ℎ < 1/2. A simple computation shows that {0 𝑓(𝑥) = { 𝑥− {

for 0 ≤ 𝑥 < 1 2

for

1 2

1 2

≤𝑥≤1

,

3.4 Rectifiable functions |

and 0 { { { 𝑓(𝑥 + ℎ) = {𝑥 + ℎ − { {1 {2

for 0 ≤ 𝑥 < 1 2

for

1 2

1 2

231

− ℎ,

− ℎ ≤ 𝑥 < 1 − ℎ,

for 1 − ℎ ≤ 𝑥 ≤ 1 .

So, the function (3.45) and its derivative have the form 0 for 0 ≤ 𝑥 < 12 − ℎ , { { { 2𝑥−1 𝑔ℎ (𝑥) = { 2ℎ + 1 for 12 − ℎ ≤ 𝑥 < 12 , { { for 12 ≤ 𝑥 ≤ 1 {1 and 0 { { {1 󸀠 𝑔ℎ (𝑥) = { ℎ { { {0

for 0 ≤ 𝑥 < for for

1 2 1 2

1 2

− ℎ,

−ℎ ≤𝑥
0 and 𝜃𝑗 ∈ [0, 2𝜋) in such a way that¹² 𝑡𝑗 − 𝑡𝑗−1 =: 𝑟𝑗 cos 𝜃𝑗 ,

𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 ) = 𝑟𝑗 sin 𝜃𝑗 .

Then 𝑡𝑗

√(𝑡𝑗 − 𝑡𝑗−1 )2 + (𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 ))2 = 𝑟𝑗 = ∫ [cos 𝜃𝑗 + 𝑓󸀠 (𝑥) sin 𝜃𝑗 ] 𝑑𝑥 , 𝑡𝑗−1

11 Observe that (3.60) may be considered as a direct analogue to formula (3.33) in Theorem 3.19 if one considers functions with values in the plane and replaces the absolute value of 𝑓󸀠 (𝑡) by the Euclidean norm of the pair (1, 𝑓󸀠 (𝑡)), see also (1.106). 12 This is possible since the function 𝑡 󳨃→ tan 𝑡 is a bijection between the interval (−𝜋/2, 𝜋/2) and the real axis.

234 | 3 Absolutely continuous functions and so 𝑡𝑗

𝑚

𝐿(𝛤(𝑓), 𝑃; [𝑎, 𝑏]) ≤ ∑ ∫ [cos 𝜃𝑗 + 𝑓󸀠 (𝑥) sin 𝜃𝑗 ] 𝑑𝑥 𝑗=1 𝑡

𝑗−1

𝑡𝑗

𝑚

𝑏

≤ ∑ ∫ √1 + 𝑓󸀠 (𝑥)2 𝑑𝑥 = ∫ √1 + 𝑓󸀠 (𝑥)2 𝑑𝑥 = 𝐼 𝑗=1 𝑡

𝑎

𝑗−1

by the Cauchy Schwarz inequality, and hence 𝐿(𝛤(𝑓); [𝑎, 𝑏]) ≤ 𝐼. To prove the converse inequality, we consider, for 𝑛 = 1, 2, 3, . . ., the equidistant partitions 𝑃𝑛 := {𝑡0,𝑛 , 𝑡1,𝑛 , . . . , 𝑡𝑛,𝑛 } given by 1 𝑛−1 (𝑏 − 𝑎), 𝑡𝑛,𝑛 := 𝑏 . 𝑡0,𝑛 := 𝑎, 𝑡1,𝑛 := 𝑎 + (𝑏 − 𝑎), . . . 𝑡𝑛−1,𝑛 := 𝑎 + 𝑛 𝑛

(3.62)

Moreover, we define piecewise constant functions 𝑔𝑛 : [𝑎, 𝑏] → ℝ by {0 𝑔𝑛 (𝑥) := { 𝑓(𝑡𝑘,𝑛 )−𝑓(𝑡𝑘−1,𝑛 ) { 𝑡𝑘,𝑛 −𝑡𝑘−1,𝑛

for 𝑥 = 𝑡𝑘,𝑛 (𝑘 = 0, 2, . . . , 𝑛) , for 𝑡𝑘−1,𝑛 < 𝑥 < 𝑡𝑘,𝑛 (𝑘 = 1, 2, . . . , 𝑛) .

From 𝑓 ∈ 𝐶1 ([𝑎, 𝑏]), it follows that 𝑔𝑛 (𝑥) → 𝑓󸀠 (𝑥) for 𝑛 → ∞ and 𝑎 ≤ 𝑥 ≤ 𝑏, and so 𝑏

𝑏

𝐼 = ∫ √1 +

𝑓󸀠 (𝑥)2

𝑎

{ } 𝑑𝑥 ≤ sup {∫ √1 + 𝑔𝑛 (𝑥)2 𝑑𝑥 : 𝑛 = 1, 2, 3, . . .} . {𝑎 }

However, by construction of 𝑔𝑛 , we see that 𝑏

∫ √1 + 𝑔𝑛 (𝑥)2 𝑑𝑥 𝑎 𝑛

2

2

= ∑ √(𝑡𝑘,𝑛 − 𝑡𝑘−1,𝑛 ) + (𝑓(𝑡𝑘,𝑛 ) − 𝑓(𝑡𝑘−1,𝑛 )) = 𝐿(𝛤(𝑓), 𝑃𝑛 ; [𝑎, 𝑏]) , 𝑘=1

and all real numbers 𝐿(𝛤(𝑓), 𝑃𝑛 ) are of course bounded above by 𝐿(𝛤(𝑓)). Therefore, we have proved the estimate 𝐼 ≤ 𝐿(𝛤(𝑓); [𝑎, 𝑏]), and so (3.60) holds. The hypothesis 𝑓 ∈ 𝐶1 ([𝑎, 𝑏]) in Proposition 3.29 is unnecessarily strong: one may show that (3.60) also holds if 𝑓 is differentiable and 𝑓󸀠 is integrable¹³, see Exercise 3.43. One might hope that formula (3.60) also holds for monotone functions which are still differentiable almost everywhere. However, this is not true:

13 Remember that we have to assume the integrability of 𝑓󸀠 since it does not follow from the differen­ tiability of 𝑓, see Example 3.15.

3.4 Rectifiable functions

| 235

Example 3.30. Let 𝜑 : [0, 1] → ℝ be the Cantor function (3.6). We claim that 𝐿(𝛤(𝜑); [0, 1]) = 2 .

(3.63)

To prove this, we cannot use formula (3.60) since 𝜑 is not everywhere differen­ tiable on [0, 1]; instead, we have to go back to Definition 3.27. Obviously, the length of any polygon with knots on the graph of 𝜑 cannot be larger than the sum of all horizontal and vertical projections of the graph of 𝜑 and all its “gaps,” which is 2. So, we trivially have 𝐿(𝛤(𝜑); [0, 1]) ≤ 2. To prove the converse estimate, we consider the polygon which starts at (0, 0), ends at (1, 1), and has as intermediate knots the endpoints of the (finitely many) intervals which we have canceled at the 𝑛-th step in the construction (3.4) of the Cantor set. The lengths of the horizontal pieces of this polygon sum up to 𝑛

2𝑘−1 2𝑛 = 1 − . 𝑘 3𝑛 𝑘=1 3 ∑

On the other hand, the oblique pieces of this polygon all have the same length, namely, 𝜆 𝑛 := √

1 1 1√ 22𝑛 + = 1 + . 22𝑛 32𝑛 2𝑛 32𝑛

Since the polygon precisely contains 2𝑛 such oblique pieces, the total length of the polygon becomes 𝐿𝑛 = 1 −

2𝑛 2𝑛 22𝑛 + 2𝑛 𝜆 𝑛 = 1 − 𝑛 + √ 1 + 2𝑛 . 𝑛 3 3 3

However, we can take the last expression arbitrarily close to 2 by choosing 𝑛 suf­ ficiently large, and so (3.63) follows. ♥ There is a general principle behind Example 3.30 which is discussed in Exercise 3.41. We point out again that we could not use (3.60) for calculating the graph length of 𝜑 in Example 3.30. Indeed, the derivative of 𝜑 at all points, where it exists, is 0, and so we would obtain the wrong value 1 for the integral if we blindly adopted formula (3.60) in this example. Our discussion shows that existence everywhere and continuity of 𝑓󸀠 is too strong, while existence a.e. and integrability of 𝑓󸀠 is too weak to ensure the equality (3.60). The following Theorem 3.31 gives the “correct” condition. Theorem 3.31. For 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]), the lower estimate 𝑏

𝐿(𝛤(𝑓); [𝑎, 𝑏]) ≥ ∫ √1 + 𝑓󸀠 (𝑥)2 𝑑𝑥 𝑎

is true, while for 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]), the equality (3.60) holds.

(3.64)

236 | 3 Absolutely continuous functions Proof. We use Proposition 3.21 which asserts that a function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]) may be represented in the form (3.38), where 𝑓ac is absolutely continuous and 𝑓sg is singular (or zero). Thus, we may write the graph (3.54) of 𝑓 in the form 𝛤(𝑓) = {(𝑥, 𝑓ac (𝑥)) + (0, 𝑓sg (𝑥)) : 𝑎 ≤ 𝑥 ≤ 𝑏} . The length of the absolutely continuous part is 𝑏

𝑏

󸀠 (𝑥)2 𝑑𝑥 = ∫ √ 1 + 𝑓󸀠 (𝑥)2 𝑑𝑥 , 𝐿 𝑎 = ∫ √1 + 𝑓ac 𝑎

(3.65)

𝑎

while the length of the singular part is simply (3.66)

𝐿 𝑠 = Var(𝑓sg ; [𝑎, 𝑏]) . Thus, the total graph length of 𝑓 adds up to 𝑏

𝐿(𝛤(𝑓); [𝑎, 𝑏]) = 𝐿 𝑎 + 𝐿 𝑠 = ∫ √1 + 𝑓󸀠 (𝑥)2 𝑑𝑥 + Var(𝑓sg ; [𝑎, 𝑏]) ,

(3.67)

𝑎

from which both statements follow. Theorem 3.31 shows that in our Example 3.30, we have intentionally chosen a function 𝜑 ∈ 𝐵𝑉([0, 1]) \ 𝐴𝐶([0, 1]) to get strict inequality in (3.64). Since the Cantor function 𝜑 is even monotone, one may also calculate the area under its graph “by hand,” see Exercise 3.12. In addition to Propositions 1.7 and 3.22, we give another result which establishes a link between a function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) and its variation function 𝑉𝑓 given in (1.13). Proposition 3.32. A function 𝑓 is rectifiable if and only if its variation function 𝑉𝑓 given in (1.13) is rectifiable; moreover, in this case, we have (3.68)

𝐿(𝛤(𝑉𝑓 ); [𝑎, 𝑏]) = 𝐿(𝛤(𝑓); [𝑎, 𝑏]) .

Proof. The first assertion follows from Theorem 1.26 (b) and the fact that the func­ tions 𝑓 ∈ 𝐵𝑉 are precisely those with a rectifiable graph. To prove (3.68), let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]). Then Proposition 3.28 (b) implies √ (𝑉𝑓 (𝑡𝑗 ) − 𝑉𝑓 (𝑡𝑗−1 ))2 + (𝑡𝑗 − 𝑡𝑗−1 )2 ≤ 𝐿 𝑓 (𝑡𝑗 ) − 𝐿 𝑓 (𝑡𝑗−1 )

(𝑗 = 1, 2, . . . , 𝑚) ,

where 𝐿 𝑓 denotes the length function (3.57) of 𝑓. Consequently, 𝑚

2

∑ √ (𝑉𝑓 (𝑡𝑗 ) − 𝑉𝑓 (𝑡𝑗−1 )) + (𝑡𝑗 − 𝑡𝑗−1 )2

𝑗=1

𝑚

≤ ∑ (𝐿 𝑓 (𝑡𝑗 ) − 𝐿 𝑓 (𝑡𝑗−1 )) = 𝐿(𝛤(𝑓), 𝑃; [𝑎, 𝑏]) , 𝑗=1

3.4 Rectifiable functions |

237

and so 𝐿(𝛤(𝑉𝑓 ); [𝑎, 𝑏]) ≤ 𝐿(𝛤(𝑓); [𝑎, 𝑏]). On the other hand, for each 𝑗 ∈ {1, 2, . . . , 𝑚}, we have, by Proposition 1.3 (c), 𝑚

2

𝐿(𝛤(𝑓), 𝑃; [𝑎, 𝑏]) = ∑ √ (𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )) + (𝑡𝑗 − 𝑡𝑗−1 )2 𝑗=1 𝑚

≤ ∑ √(𝑉𝑓 (𝑡𝑗 ) − 𝑉𝑓 (𝑡𝑗−1 )) + (𝑡𝑗 − 𝑡𝑗−1 )2 ≤ 𝐿(𝛤(𝑉𝑓 ), 𝑃; [𝑎, 𝑏]) , 𝑗=1

which proves the converse estimate. A certain refinement of Proposition 3.32 is given in Exercise 3.44. We collect in the following Table 3.3 the properties of a function 𝑓 : [𝑎, 𝑏] → ℝ which carry over to its variation function, or vice versa. Table 3.3. Relations between 𝑓 and 𝑉𝑓 . 𝑓 ∈ 𝐶([𝑎, 𝑏]) 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) 𝑓 ∈ 𝐿𝑖𝑝([𝑎, 𝑏]) 𝑓 ∈ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) 𝑓 differentiable a.e. 𝑓 rectifiable

⇔ ⇔ ⇔ ⇐ ⇔ ⇔ ⇔

𝑉𝑓 ∈ 𝐶([𝑎, 𝑏]) 𝑉𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) 𝑉𝑓 ∈ 𝐿𝑖𝑝([𝑎, 𝑏]) 𝑉𝑓 ∈ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) 𝑉𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) 𝑉𝑓 differentiable a.e. 𝑉𝑓 rectifiable

(Theorem 1.26 (a)) (Theorem 1.26 (b)) (Theorem 1.26 (c)) (Theorem 1.26 (d)) (Theorem 1.26 (e)) (Proposition 3.22) (Proposition 3.32)

At this point, it is time to take a breath and to recall all the estimates for functions of bounded variation which, as a rule, turn into equalities for absolutely continuous functions. – The estimate 𝑥

𝑉𝑓 (𝑥) ≥ ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 𝑎

–

for the variation function (1.13) holds for every function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), with equal­ ity in case 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]). The estimate 𝑏

Var(𝑓; [𝑎, 𝑏]) ≥ ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 𝑎

for the total variation (1.4) holds for every function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), with equality¹⁴ in case 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]).

14 Taking into account the definition (1.16) of the 𝐵𝑉-norm, we even see that ‖𝑓‖𝐵𝑉 = ‖𝑓󸀠 ‖𝐿 1 for 𝑓 ∈ 𝐴𝐶𝑜 ([𝑎, 𝑏]) = {𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]), 𝑓(𝑎) = 0}.

238 | 3 Absolutely continuous functions –

The estimate

𝑥

𝐿 𝑓 (𝑥) ≥ ∫ √1 + 𝑓󸀠 (𝑡)2 𝑑𝑡 𝑎

–

for the length function (3.57) holds for every function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), with equality in case 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]). The estimate 𝑏

𝐿(𝛤(𝑓); [𝑎, 𝑏]) ≥ ∫ √1 + 𝑓󸀠 (𝑡)2 𝑑𝑡 𝑎

–

for the graph length (3.56) holds for every function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), with equality in case 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]). The estimate 𝑏

𝑓(𝑏) − 𝑓(𝑎) ≥ ∫ 𝑓󸀠 (𝑡) 𝑑𝑡 𝑎

–

holds for every increasing function 𝑓 : [𝑎, 𝑏] → ℝ, with equality in case 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]). The equality¹⁵ 𝑙𝑖𝑝(𝑓; [𝑎, 𝑏]) = esssup {|𝑓󸀠 (𝑡)| : 𝑎 ≤ 𝑡 ≤ 𝑏} for the smallest Lipschitz constant (0.68) holds for 𝑓 ∈ 𝐿𝑖𝑝([𝑎, 𝑏]).

The monotonically increasing continuous Cantor function 𝜑 which is not absolutely continuous may be used throughout as an example for strict inequalities in this list. The reason for this is that 𝜑󸀠 (𝑥) = 0 a.e. on [0, 1], although 𝜑 is not constant. For abso­ lutely continuous functions, this is impossible: Proposition 3.33. If 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) satisfies 𝑓󸀠 (𝑥) = 0 a.e. on [𝑎, 𝑏], then 𝑓 is constant. Proof. Given a fixed 𝑐 ∈ (𝑎, 𝑏], we show that 𝑓(𝑐) = 𝑓(𝑎). Let 𝑀 := {𝑥 : 𝑎 ≤ 𝑥 ≤ 𝑐, 𝑓󸀠 (𝑥) = 0}. Since 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]), for 𝜀 > 0, we find 𝛿 > 0 such that 𝛤(𝑓; 𝑆) ≤ 𝜀 for any 𝑆 ∈ 𝛴([𝑎, 𝑏]) satisfying 𝛩(𝑆) ≤ 𝛿. Furthermore, for fixed 𝑥 ∈ 𝑀, we may choose, by definition of 𝑀, a small interval [𝑎𝑥 , 𝑐𝑥 ] ⊂ [𝑎, 𝑐] such that 𝑎𝑥 ≤ 𝑥 ≤ 𝑐𝑥 ,

|𝑓(𝑐𝑥 ) − 𝑓(𝑎𝑥 )| ≤ 𝜀(𝑐𝑥 − 𝑎𝑥 ) .

Now, by the Vitali covering theorem, we can find a finite collection 𝑆 := {[𝑎𝑥,1 , 𝑐𝑥,1 ], . . . , [𝑎𝑥,𝑛 , 𝑐𝑥,𝑛 ]} of nonoverlapping intervals of this sort such that 𝜆 (𝑀 \ [𝑎𝑥,1 , 𝑐𝑥,1 ] ∪ . . . ∪ [𝑎𝑥,𝑛 , 𝑐𝑥,𝑛 ]) ≤ 𝛿 .

15 Taking into account the definition (0.70) of the 𝐿𝑖𝑝-norm, we even see that ‖𝑓‖𝐿𝑖𝑝 = ‖𝑓󸀠 ‖𝐿 ∞ for 𝑓 ∈ 𝐿𝑖𝑝𝑜 ([𝑎, 𝑏]) = {𝑓 ∈ 𝐿𝑖𝑝([𝑎, 𝑏]), 𝑓(𝑎) = 0}.

3.4 Rectifiable functions

| 239

Since 𝜆([𝑎, 𝑐] \ 𝑀) = 0, by assumption, we have 𝜆 ([𝑎, 𝑐] \ [𝑎𝑥,1 , 𝑐𝑥,1 ] ∪ . . . ∪ [𝑎𝑥,𝑛 , 𝑐𝑥,𝑛 ]) = 𝜆 ([𝑎, 𝑐] \ [𝑎𝑥,1 , 𝑐𝑥,1 ] ∪ . . . ∪ [𝑎𝑥,𝑛 , 𝑐𝑥,𝑛 ]) ≤ 𝛿 . Suppose, without loss of generality, that 𝑎 ≤ 𝑎𝑥,1 < 𝑐𝑥,1 ≤ 𝑎𝑥,2 < . . . < 𝑐𝑥,𝑛−1 ≤ 𝑎𝑥,𝑛 < 𝑐𝑥,𝑛 ≤ 𝑐, and put 𝑐𝑥,0 := 𝑎 and 𝑎𝑥,𝑛+1 := 𝑐. Then 𝑛

𝑛

𝛩(𝑆 ∪ {𝑎, 𝑐}) = ∑ (𝑎𝑥,𝑘+1 − 𝑐𝑥,𝑘 ) = 𝜆 ([𝑎, 𝑐] \ ⋃[𝑎𝑥,𝑘 , 𝑐𝑥,𝑘 ]) ≤ 𝛿 , 𝑘=0

and hence

𝑘=1 𝑛

∑ |𝑓(𝑎𝑥,𝑘+1 ) − 𝑓(𝑐𝑥,𝑘 )| ≤ 𝜀 . 𝑘=0

Furthermore, the definition of the intervals [𝑎𝑥,𝑘 , 𝑐𝑥,𝑘 ] implies that 𝑛

𝑛

∑ |𝑓(𝑐𝑥,𝑘 ) − 𝑓(𝑎𝑥,𝑘 )| ≤ 𝜀 ∑ (𝑐𝑥,𝑘 ) − 𝑎𝑥,𝑘 ) ≤ 𝜀(𝑐 − 𝑎) .

𝑘=1

𝑘=1

Combining these estimates, we obtain 𝑛

𝑛

|𝑓(𝑐) − 𝑓(𝑎)| ≤ ∑ |𝑓(𝑎𝑥,𝑘+1 ) − 𝑓(𝑐𝑥,𝑘 )| + ∑ |𝑓(𝑐𝑥,𝑘 ) − 𝑓(𝑎𝑥,𝑘 )| ≤ 𝜀(𝑐 − 𝑎 + 1) . 𝑘=0

𝑘=1

Since 𝜀 > 0 was arbitrary, we conclude that 𝑓(𝑐) = 𝑓(𝑎) as claimed. Proposition 3.33 shows that a singular function in the sense of Definition 3.7 cannot be absolutely continuous.¹⁶ So Theorem 3.9 implies that a singular function must fail to have the Luzin property. The Cantor function (3.6) is of course a standard example. In general, we may formulate a “Golden Rule” which states that whenever one wants to see whether or not a result for absolutely continuous functions carries over to func­ tions of bounded variation, it is a useful device to take the Cantor function (3.6) as a “test animal.” Observe that we could have deduced Proposition 3.33 also very easily from The­ orem 3.18: indeed, our hypothesis implies that the integral in (3.18) is zero, and so 𝑓(𝑥) ≡ 𝑓(𝑎). We have given an independent proof which gives some insight and uses an interesting compactness argument. Conversely, we also note that Proposition 3.33 may be used to prove the equality (3.18) for 𝑓 ∈ 𝐴𝐶 quite easily once we know that 𝑓󸀠 ∈ 𝐿 1 . In fact, the function 𝑓 ̃ : [𝑎, 𝑏] → ℝ defined by 𝑥

̃ := 𝑓(𝑎) + ∫ 𝑓󸀠 (𝑡) 𝑑𝑡 𝑓(𝑥)

(𝑎 ≤ 𝑥 ≤ 𝑏)

𝑎

16 It is precisely this fact which shows that the decomposition (3.38) makes sense.

240 | 3 Absolutely continuous functions is absolutely continuous, by Proposition 3.4, and satisfies 𝑓󸀠̃ = 𝑓󸀠 a.e. on [𝑎, 𝑏]. So, ̃ from Proposition 3.33, we conclude that 𝑓 ̃ − 𝑓 is constant on [𝑎, 𝑏]. However, 𝑓(𝑎) = 𝑓(𝑎), and (3.18) follows. In this way, we have obtained a complete answer to the two questions raised at the beginning of Section 3.3.

3.5 The Riesz–Medvedev theorem As we have seen in Section 3.3, we may characterize Lipschitz continuous and abso­ lutely continuous functions through properties of their derivatives in the following way: – Lipschitz continuous functions are precisely those with 𝐿 ∞ -derivatives a.e. (The­ orem 3.20). – Absolutely continuous functions are precisely those with 𝐿 1 -derivatives a.e. (Proposition 3.4 and Theorem 3.18). Now, the question arises if one may also characterize the functions with 𝐿 𝑝 -deriva­ tives for 1 < 𝑝 < ∞; by the inclusions (0.15), such functions should be intermediate between Lipschitz continuous and absolutely continuous functions. It turns out that in this way, we get nothing else but the space 𝑅𝐵𝑉𝑝 which, by (2.93), is in fact interme­ diate between the spaces 𝐿𝑖𝑝 and 𝐴𝐶. This is the contents of the famous Riesz theorem which reads as follows. Theorem 3.34 (Riesz). Let 1 < 𝑝 < ∞. Then a function 𝑓 belongs to 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) if and only if 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) and 𝑓󸀠 ∈ 𝐿 𝑝 ([𝑎, 𝑏]). Moreover, in this case, the equality 𝑏

Var𝑅𝑝 (𝑓; [𝑎, 𝑏])

=

𝑝 ‖𝑓󸀠 ‖𝐿 𝑝

= ∫ |𝑓󸀠 (𝑥)|𝑝 𝑑𝑥

(3.69)

𝑎

holds, where Var𝑅𝑝 (𝑓; [𝑎, 𝑏]) denotes the 𝑝-variation (2.88) of 𝑓 in Riesz’s sense. Proof. To prove the “if” part, we suppose that 𝑓 is absolutely continuous with 𝑓󸀠 ∈ 𝐿 𝑝 ([𝑎, 𝑏]). For 𝑥, 𝑦 ∈ [𝑎, 𝑏] with 𝑥 < 𝑦, we then get, by Hölder’s inequality (0.14), 𝑝 󵄨󵄨 𝑦 󵄨󵄨𝑝 𝑦 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󸀠 󸀠 |𝑓(𝑦) − 𝑓(𝑥)| = 󵄨󵄨∫ 𝑓 (𝑡) 𝑑𝑡󵄨󵄨 ≤ (∫ |𝑓 (𝑡)| 𝑑𝑡) 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑥 󵄨󵄨 𝑥 𝑝

𝑦

[ ≤ [(∫ 𝑑𝑡) [ Consequently,

𝑥

𝑝

1−1/𝑝

𝑦

𝑦

] 󸀠 𝑝 𝑝−1 󸀠 𝑝 ] ∫ |𝑓 (𝑡)| 𝑑𝑡 = (𝑦 − 𝑥) ∫ |𝑓 (𝑡)| 𝑑𝑡 . ]

𝑥

𝑥

𝑦

𝑏

𝑥

𝑎

|𝑓(𝑥) − 𝑓(𝑦)|𝑝 ≤ ∫ |𝑓󸀠 (𝑡)|𝑝 𝑑𝑡 ≤ ∫ |𝑓󸀠 (𝑡)|𝑝 𝑑𝑡 , |𝑥 − 𝑦|𝑝−1

3.5 The Riesz–Medvedev theorem |

241

𝑝

and hence Var𝑅𝑝 (𝑓; [𝑎, 𝑏]) ≤ ‖𝑓󸀠 ‖𝐿 𝑝 , which shows that 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) and proves one estimate for (3.69). We already know from Proposition 2.52 that the inclusion 𝑅𝐵𝑉 𝑝 ([𝑎, 𝑏]) ⊆ 𝐴𝐶([𝑎, 𝑏]) is true for 𝑝 > 1, and so 𝑓󸀠 (𝑥) exists a.e. on [𝑎, 𝑏]. Consequently, to prove the “only if” part, we merely have to show that 𝑓󸀠 ∈ 𝐿 𝑝 ([𝑎, 𝑏]). As in the proof of Proposition 3.29, for every 𝑛 ∈ ℕ, we consider the equidis­ tant partition (3.62), and define piecewise constant functions 𝑔𝑛 : [𝑎, 𝑏] → ℝ for 𝑛 = 1, 2, 3, . . . and 𝑘 = 0, 1, . . . , 𝑛 by 𝑓(𝑡

)−𝑓(𝑡 )

𝑘,𝑛 for 𝑡𝑘,𝑛 ≤ 𝑡 < 𝑡𝑘+1,𝑛 , { 𝑘+1,𝑛 𝑔𝑛 (𝑡) := { 𝑡𝑘+1,𝑛 −𝑡𝑘,𝑛 0 for 𝑡 = 𝑏 . { A somewhat cumbersome, but straightforward calculation, then shows that 𝑔𝑛 (𝑥) → 𝑓󸀠 (𝑥) a.e. in [𝑎, 𝑏]. From Fatou’s theorem (Theorem 0.5), we conclude that

𝑏

𝑏

|𝑓(𝑡𝑘+1,𝑛 ) − 𝑓(𝑡𝑘,𝑛 )|𝑝 ≤ Var𝑅𝑝 (𝑓; [𝑎, 𝑏]) . 𝑝−1 |𝑡 − 𝑡 | 𝑘+1,𝑛 𝑘,𝑛 𝑘=1 𝑛

∫ |𝑓󸀠 (𝑡)|𝑝 𝑑𝑡 ≤ lim ∫ |𝑔𝑛 (𝑡)|𝑝 𝑑𝑡 = lim ∑ 𝑎

𝑛→∞

𝑛→∞

𝑎

This shows that 𝑓󸀠 ∈ 𝐿 𝑝 ([𝑎, 𝑏]) and proves the other estimate for (3.69). We make some comments on Theorem 3.34. First of all, we point out that an analogous result is not true in case 𝑝 = 1 since functions 𝑓 ∈ 𝑅𝐵𝑉1 = 𝐵𝑉 are generally not contin­ uous, let alone absolutely continuous. However, a certain analogue of Theorem 3.34 for 𝑝 = ∞ is true: since 𝑅𝐵𝑉∞ = 𝐿𝑖𝑝, Theorem 3.20 may be considered as such an analogue. Moreover, Theorem 3.34 may be used to give another proof of Proposition 2.54. In fact, given 𝑓 ∈ 𝑅𝐵𝑉𝑞 ([𝑎, 𝑏]), we have 𝑓󸀠 ∈ 𝐿 𝑞 ([𝑎, 𝑏]), and so 𝑓󸀠 ∈ 𝐿 𝑝 ([𝑎, 𝑏]). Therefore, 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) for any 𝑝 ≤ 𝑞. Another interesting consequence of Theorem 3.34 is the following. If we define the subspace 𝑅𝐵𝑉𝑜𝑝 , as in Proposition 2.51, by 𝑅𝐵𝑉𝑜𝑝 ([𝑎, 𝑏]) := {𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) : 𝑓(𝑎) = 0} ,

(3.70)

the differential operator 𝐷 defined by 𝐷𝑓 := 𝑓󸀠 is a linear surjective isometry between 𝑅𝐵𝑉𝑜𝑝 ([𝑎, 𝑏]) and 𝐿 𝑝 ([𝑎, 𝑏]). This is not only of theoretical interest, but also of practical use: it makes it possible to calculate the norm of functions 𝑓 ∈ 𝑅𝐵𝑉𝑝 quite easily. Here is a simple example. Example 3.35. For 0 < 𝜏 < 1, let 𝑓𝜏 : [0, 1] → ℝ be defined as in Example 2.78. Then 𝑓𝜏󸀠 exists and belongs to 𝐿 𝑝 ([0, 1]) if and only if 𝑝(1 − 𝜏) < 1, which reconfirms (2.153). Moreover, in this case, we have 1 𝑝 ‖𝑓𝜏󸀠 ‖𝐿 𝑝

= ∫ 𝜏𝑝 𝑥𝑝(𝜏−1) 𝑑𝑥 = 0

𝜏𝑝 , 1 − (1 − 𝜏)𝑝

in accordance with our result for ‖𝑓𝜏 ‖𝑅𝐵𝑉𝑝 in Example 2.78.

♥

242 | 3 Absolutely continuous functions The meaning of Theorem 3.34 becomes clearer if we write the Riesz variation (2.87) with respect to 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) in the form 𝑚

Var𝑅𝑝 (𝑓, 𝑃; [𝑎, 𝑏]) = ∑ ( 𝑗=1

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| 𝑡𝑗 − 𝑡𝑗−1

𝑝

(3.71)

) (𝑡𝑗 − 𝑡𝑗−1 ) .

Indeed, the expression (3.71) may then be viewed as a Riemann sum, and for 𝜇(𝑃) → 0, see (1.2), the summation becomes an integral, while the term after the summation approaches, roughly speaking, |𝑓󸀠 (𝑡)|𝑝 𝑑𝑡. The notation (3.71) also moti­ vates the definition (2.96) for the Riesz variation with respect to an arbitrary Young function 𝜙. In this more general setting, the expression after the summation in (2.96) approaches 𝜙(|𝑓󸀠 (𝑡)|) 𝑑𝑡 as 𝜇(𝑃) → 0. Therefore, the following important generaliza­ tion of Theorem 3.34 which involves the Orlicz class L𝜙 ([𝑎, 𝑏]) ( Definition 0.16) and is due to Medvedev [211] appears very natural. Theorem 3.36 (Medvedev). Let 𝜙 be a Young function which satisfies condition ∞1 , see (2.16). Then a function 𝑓 belongs to 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) if and only if 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) and 𝑓󸀠 ∈ L𝜙 ([𝑎, 𝑏]). Moreover, in this case, the equality 𝑏

Var𝑅𝜙 (𝑓; [𝑎, 𝑏]) = ∫ 𝜙(|𝑓󸀠 (𝑥)|) 𝑑𝑥

(3.72)

𝑎

holds, where Var𝑅𝜙 (𝑓; [𝑎, 𝑏]) denotes the 𝜙-variation (2.97) of 𝑓 in Riesz’s sense. Proof. Suppose first that 𝑓 ∈ 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]), and fix 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏]). Let 𝜀 > 0. From our hypothesis 𝜙 ∈ ∞1 , it follows that for each 𝜀 > 0 and 𝑘 > 0, we can find a 𝛿 > 0 such that 0 ≤ 𝑐 ≤ 𝛿 implies 𝑘 𝑐𝜙−1 ( ) ≤ 𝜀 . 𝑐 Taking, in particular, 𝑐 := 𝛩(𝑆) and 𝑘 := Var𝑅𝜙 (𝑓; [𝑎, 𝑏]), we deduce that 𝛩(𝑆) ≤ 𝛿 which implies Var𝑅𝜙 (𝑓; [𝑎, 𝑏]) ) ≤ 𝜀. (3.73) 𝛩(𝑆)𝜙−1 ( 𝛩(𝑆) Now, applying the discrete Jensen inequality (2.18) to 𝛼𝑘 := 𝑏𝑘 − 𝑎𝑘 ,

𝑢𝑘 :=

|𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| 𝑏𝑘 − 𝑎𝑘

(𝑘 = 1, 2, . . . , 𝑛)

and using the notation (3.1), we get the estimates ∑𝑛𝑘=1 (𝑏𝑘 − 𝑎𝑘 )𝜙 ( 𝛤(𝑓; 𝑆) )≤ 𝜙( 𝛩(𝑆) 𝛩(𝑆)

|𝑓(𝑏𝑘 )−𝑓(𝑎𝑘 )| ) 𝑏𝑘 −𝑎𝑘

=

Var𝑅𝜙 (𝑓, 𝑆; [𝑎, 𝑏]) 𝛩(𝑆)

≤

Var𝑅𝜙 (𝑓; [𝑎, 𝑏]) 𝛩(𝑆)

Combining this with (3.73) and using the monotonicity of 𝜙−1 , we obtain 𝑅

𝛤(𝑓; 𝑆) = 𝛩(𝑆)

Var𝜙 (𝑓; [𝑎, 𝑏]) 𝛤(𝑓; 𝑆) ≤ 𝛩(𝑆)𝜙−1 ( ) ≤ 𝜀. 𝛩(𝑆) 𝛩(𝑆)

.

3.5 The Riesz–Medvedev theorem | 243

This shows that 𝑓 is absolutely continuous on [𝑎, 𝑏]. By Theorem 3.18, 𝑓 admits a representation 𝑥

𝑓(𝑥) = ∫ 𝑔(𝑡) 𝑑𝑡

(𝑎 ≤ 𝑥 ≤ 𝑏)

𝑎

with a suitable function 𝑔 ∈ 𝐿 1 ([𝑎, 𝑏]) satisfying 𝑓󸀠 = 𝑔 a.e. on [𝑎, 𝑏]. It remains to show that 𝑏

(3.74)

∫ 𝜙(|𝑔(𝑡)|) 𝑑𝑡 < ∞ . 𝑎

For 𝑛 ∈ ℕ, consider the equidistant partition 𝑃𝑛 := {𝑡0,𝑛 , 𝑡1,𝑛 , . . . , 𝑡𝑛−1,𝑛 , 𝑡𝑛,𝑛 } ∈ P([𝑎, 𝑏]) given by (3.62), and define functions 𝑔𝑛 : [𝑎, 𝑏] → ℝ as in the proof of Propo­ sition 3.29. Then the sequence (𝑔𝑛 )𝑛 converges a.e. on [𝑎, 𝑏] to 𝑔, and therefore 𝑏

𝑏

{ } ∫ 𝜙(|𝑔(𝑡)|) 𝑑𝑡 ≤ sup {∫ 𝜙(|𝑔𝑛 (𝑡)|) 𝑑𝑡 : 𝑛 = 1, 2, 3, . . .} , 𝑎 {𝑎 } by Fatou’s theorem (Theorem 0.5). However, 𝑏

𝑡𝑘,𝑛

𝑛

∫ 𝜙(|𝑔(𝑡)|) 𝑑𝑡 = ∑ ∫ 𝜙 ( 𝑘=1 𝑡

𝑎

𝑘−1,𝑛

𝑛

= ∑ 𝜙( 𝑘=1

|𝑓(𝑡𝑘,𝑛 ) − 𝑓(𝑡𝑘−1,𝑛 )| ) 𝑑𝑡 𝑡𝑘,𝑛 − 𝑡𝑘−1,𝑛

|𝑓(𝑡𝑘,𝑛 ) − 𝑓(𝑡𝑘−1,𝑛 )| ) (𝑡𝑘,𝑛 − 𝑡𝑘−1,𝑛 ) 𝑡𝑘,𝑛 − 𝑡𝑘−1,𝑛

= Var𝑅𝜙 (𝑓, 𝑃𝑛 ; [𝑎, 𝑏]) ≤ Var𝑅𝜙 (𝑓; [𝑎, 𝑏]), not only proving (3.74), but also the estimate ≥ in (3.72). Conversely, suppose now that 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) and 𝑔 := 𝑓󸀠 satisfies (3.74). Fix a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑛 } ∈ P([𝑎, 𝑏]). Then 𝑡𝑘

|𝑓(𝑡𝑘 ) − 𝑓(𝑡𝑘−1 )| 1 𝜙( ) = 𝜙( ∫ 𝑔(𝑡) 𝑑𝑡) . 𝑡𝑘 − 𝑡𝑘−1 𝑡𝑘 − 𝑡𝑘−1 𝑡𝑘−1

Applying to the right-hand side the continuous Jensen inequality (2.19) for 𝛼(𝑡) ≡ 1, 𝑢(𝑡) := |𝑔(𝑡)| and [𝑐, 𝑑] := [𝑡𝑘−1 , 𝑡𝑘 ], we get the estimate 𝑡𝑘

𝑡𝑘

𝑡𝑘−1

𝑡𝑘−1

1 1 ∫ 𝑔(𝑡) 𝑑𝑡) ≤ ∫ 𝜙(|𝑔(𝑡)|) 𝑑𝑡 . 𝜙( 𝑡𝑘 − 𝑡𝑘−1 𝑡𝑘 − 𝑡𝑘−1 Multiplying by 𝑡𝑘 − 𝑡𝑘−1 and summing up over 𝑘 = 1, 2, . . . , 𝑛 gives Var𝑅𝜙 (𝑓, 𝑃; [𝑎, 𝑏])

𝑛

𝑏

|𝑓(𝑡𝑘 ) − 𝑓(𝑡𝑘−1 )| = ∑ 𝜙( ) (𝑡𝑘 − 𝑡𝑘−1 ) ≤ ∫ 𝜙(|𝑔(𝑡)|) 𝑑𝑡 . 𝑡𝑘 − 𝑡𝑘−1 𝑘=1 𝑎

244 | 3 Absolutely continuous functions Since the last term in this estimate is independent of 𝑃, we have proved not only that 𝑓 ∈ 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]), but also the estimate ≤ in (3.72). Clearly, for 𝜙(𝑡) = 𝑡𝑝 with 1 < 𝑝 < ∞, we have 𝜑 ∈ ∞1 , and (3.72) reduces to (3.69). Thus, Theorem 3.36 contains Theorem 3.34 as a special case. Moreover, since Theo­ rem 3.34 is false for 𝜙(𝑡) = 𝑡, i.e. 𝑅𝐵𝑉𝜙 = 𝐵𝑉, we cannot drop the assumption 𝜙 ∈ ∞1 in Theorem 3.36.

3.6 Higher order Riesz-type variations In this section, we use the higher order divided differences considered in Defini­ tion 2.71 to introduce and study higher order variations in Riesz’s sense. The following Definitions 3.37 and 3.38 are parallel to Definitions 2.71 and 2.72. Definition 3.37. Given a function 𝑓 : [𝑎, 𝑏] → ℝ and points 𝑡0 , 𝑡1 , . . . , 𝑡𝑘 ∈ [𝑎, 𝑏], let the higher order divided differences be defined as in (2.139). Given a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), 𝑘 ∈ ℕ, and 𝑝 ∈ [1, ∞), we set 𝑚−𝑘+1

Var𝑅𝑘,𝑝 (𝑓, 𝑃; [𝑎, 𝑏]) := ∑

𝑗=1

|𝑓[𝑡𝑗 , . . . , 𝑡𝑗+𝑘−1 ] − 𝑓[𝑡𝑗−1 , . . . , 𝑡𝑗+𝑘−2 ]|𝑝 |𝑡𝑗 − 𝑡𝑗−1 |𝑝−1

(3.75)

and call (3.75) the (𝑘, 𝑝)-variation of 𝑓 with respect to 𝑃 (in Riesz’s sense) on [𝑎, 𝑏]. ◼ Again, Definition 3.37 contains some notions of variation defined previously in the literature. In the special case 𝑘 = 1 and 𝑝 = 1, (3.75) reduces to Jordan’s classical definition of Var(𝑓, 𝑃; [𝑎, 𝑏]), while in the special case 𝑘 = 1 and 𝑝 > 1, (3.75) reduces to Riesz’s variation Var𝑅𝑝 (𝑓, 𝑃; [𝑎, 𝑏]) which we studied in detail in Section 2.4. On the other hand, for arbitrary 𝑘 ∈ ℕ and 𝑝 = 1, the (𝑘, 1)-variation Var𝑅𝑘,1 (𝑓, 𝑃; [𝑎, 𝑏]) coincides with the (𝑘, 1)-variation Var𝑊 𝑘,1 (𝑓, 𝑃; [𝑎, 𝑏]) in Popoviciu’s sense, see (2.142). However, for 𝑘 ∈ ℕ and 𝑝 > 1, the expression (3.75) is of course different from the (𝑘, 𝑝)-variation Var𝑊 𝑘,𝑝 (𝑓, 𝑃; [𝑎, 𝑏]) considered in (2.140) since we have the additional denominator |𝑡𝑗 − 𝑡𝑗−1 |𝑝−1 . Definition 3.38. Given 𝑘 ∈ ℕ and 𝑝 ∈ [1, ∞), we call the (possibly infinite) number Var𝑅𝑘,𝑝 (𝑓; [𝑎, 𝑏]) := sup {Var𝑅𝑘,𝑝 (𝑓, 𝑃; [𝑎, 𝑏]) : 𝑃 ∈ P([𝑎, 𝑏])} ,

(3.76)

where the supremum is taken over all partitions of [𝑎, 𝑏], the total (𝑘, 𝑝)-variation of 𝑓 (in Riesz’s sense) on [𝑎, 𝑏]. The set 𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]) := {𝑓 ∈ 𝐵([𝑎, 𝑏]) : Var𝑅𝑘,𝑝 (𝑓; [𝑎, 𝑏]) < ∞} is called the space of all functions of bounded (𝑘, 𝑝)-variation (in Riesz’s sense) on [𝑎, 𝑏]. ◼

3.6 Higher order Riesz-type variations

| 245

By what we observed before, we have the special cases 𝑅𝐵𝑉1,𝑝 ([𝑎, 𝑏]) = 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]),

𝑅𝐵𝑉𝑘,1 ([𝑎, 𝑏]) = 𝑊𝐵𝑉𝑘,1 ([𝑎, 𝑏]) .

In rather the same way as we have done this for the space 𝑅𝐵𝑉 𝑝 in Proposition 2.51, one may show that the linear space 𝑅𝐵𝑉𝑘,𝑝 with norm 𝑘−1

‖𝑓‖𝑅𝐵𝑉𝑘,𝑝 := ∑ |𝑓(𝑖) (𝑎)| + Var𝑅𝑘,𝑝 (𝑓; [𝑎, 𝑏])1/𝑝

(3.77)

𝑖=0

is a Banach space. One may also show (Exercise 3.49) that 𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]) is “decreas­ ing” with respect to 𝑘 ∈ ℕ and “increasing” with respect to 𝑝 ∈ [1, ∞). The spaces 𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]) have particularly interesting properties for 𝑝 > 1. As we have shown for 𝑘 = 1 in Theorem 3.34, a function 𝑓 : [𝑎, 𝑏] → ℝ belongs to 𝑅𝐵𝑉1,𝑝 ([𝑎, 𝑏]) = 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) for 1 < 𝑝 < ∞ if and only if 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) and 𝑓󸀠 ∈ 𝐿 𝑝 ([𝑎, 𝑏]); in this case 𝑏

Var𝑅𝑝 (𝑓; [𝑎, 𝑏])

= ∫ |𝑓󸀠 (𝑡)|𝑝 𝑑𝑡 .

(3.78)

𝑎

To formulate a parallel result for functions in 𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]), we denote by 𝐴𝐶 ([𝑎, 𝑏]) the linear space of all functions 𝑓 : [𝑎, 𝑏] → ℝ whose 𝑛-th derivative 𝑓(𝑛) exists and is absolutely continuous on [𝑎, 𝑏], equipped with the norm¹⁷ 𝑛

𝑛−1

(𝑗)

𝑏

𝑛

(𝑛)

‖𝑓‖𝐴𝐶𝑛 = ∑ |𝑓 (𝑎)| + ‖𝑓 ‖𝐴𝐶 = ∑ |𝑓 (𝑎)| + ∫ |𝑓(𝑛+1) (𝑡)| 𝑑𝑡 . 𝑗=0

(𝑗)

𝑗=0

(3.79)

𝑎

The following is a generalization of Theorem 3.34 to higher order Riesz spaces. Theorem 3.39. Let 1 < 𝑝 < ∞ and 𝑘 ∈ ℕ. Then a function 𝑓 belongs to 𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]) if and only if 𝑓 ∈ 𝐴𝐶𝑘−1 ([𝑎, 𝑏]) and 𝑓(𝑘) ∈ 𝐿 𝑝 ([𝑎, 𝑏]). Moreover, in this case, the equality 𝑝

Var𝑅𝑘,𝑝 (𝑓; [𝑎, 𝑏])

‖𝑓(𝑘) ‖𝐿 𝑝

𝑏

𝑝

|𝑓(𝑘) (𝑡)| ) 𝑑𝑡 = = ∫( 𝑝 ((𝑘 − 1)!) (𝑘 − 1)!

(3.80)

𝑎

holds, where Var𝑅𝑘,𝑝 (𝑓; [𝑎, 𝑏]) denotes the (𝑘, 𝑝)-variation (3.76) of 𝑓 in Riesz’s sense. Proof. For 𝑘 = 1, the assertion of Theorem 3.39 is just Riesz’s theorem (Theorem 3.34), so we may assume that 𝑘 ≥ 2. Following [229], we use a number of auxiliary results whose proofs are left to the reader as exercises. Suppose first that 𝑓 : [𝑎, 𝑏] → ℝ belongs to 𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]). By Exercise 2.46, we know that then the derivative 𝑓(𝑘−1) exists on [𝑎, 𝑏].

17 This is of course a special case of the general construction of 𝑋𝑛 considered in (0.43). The special case 𝑛 = 1 is considered in Exercise 3.53.

246 | 3 Absolutely continuous functions Let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) be a partition of [𝑎, 𝑏]. For any subinterval [𝑡𝑗 , 𝑡𝑗+1 ], we define a partition 𝑃𝑗 ∈ P([𝑡𝑗 , 𝑡𝑗+1 ]) in the form 𝑡𝑗 = 𝑡𝑗,1 < 𝑡𝑗,2 < . . . < 𝑡𝑗,𝑘−1 < 𝑡𝑗,𝑘 = 𝑡𝑗 + ℎ < . . .

(3.81)

. . . < 𝑡𝑗,𝑘+1 = 𝑡𝑗+1 − ℎ < 𝑡𝑗,𝑘+2 < . . . < 𝑡𝑗,2𝑘 = 𝑡𝑗+1 , where ℎ := min {

𝑡𝑗+1 − 𝑡𝑗 2

: 𝑗 = 0, 1, 2, . . . , 𝑚 − 1} .

Using the shortcut |𝑓[𝑡𝑗+1 − ℎ, 𝑡𝑗,𝑘+2 , . . . , 𝑡𝑗,2𝑘−1 ] − 𝑓[𝑡𝑗 , 𝑡𝑗,2 , . . . , 𝑡𝑗,𝑘−1 , 𝑡𝑗 + ℎ]|𝑝

𝐹𝑗,𝑘 (𝑃𝑗 ; ℎ) :=

|𝑡𝑗+1 − 𝑡𝑗 |𝑝−1

,

by Exercise 3.48, the derivative 𝑓(𝑘−1) is continuous, and so |𝑓(𝑘−1) (𝑡𝑗+1 ) − 𝑓(𝑘−1) (𝑡𝑗 )|𝑝 (𝑡𝑗+1 − 𝑡𝑗 )𝑝−1

= lim

lim

lim

ℎ→0+ 𝑡𝑗,𝑘−1 →𝑡𝑗 𝑡𝑗,𝑘+2 →𝑡𝑗+1

𝐹𝑗,𝑘 (𝑃𝑗 ; ℎ) .

Consequently, 𝑚

∑

|𝑓(𝑘−1) (𝑡𝑗+1 ) − 𝑓(𝑘−1) (𝑡𝑗 )|𝑝 |𝑡𝑗+1 − 𝑡𝑗 |𝑝−1

𝑗=0 𝑚

= ∑ lim 𝑗=0

lim

lim

ℎ→0+ 𝑡𝑗,𝑘−1 →𝑡𝑗 𝑡𝑗,𝑘+2 →𝑡𝑗+1

𝐹𝑗,𝑘 (𝑃𝑗 ; ℎ) ≤ Var𝑅𝑘,𝑝 (𝑓; [𝑎, 𝑏]) .

Thus, from Theorem 3.34, we conclude that 𝑓(𝑘−1) ∈ 𝐴𝐶([𝑎, 𝑏]) and 𝑓(𝑘) ∈ 𝐿 𝑝 ([𝑎, 𝑏]). Moreover, 𝑏

∫ 𝑎

|𝑓(𝑘) (𝑡)|𝑝 𝑑𝑡 ≤ Var𝑅𝑘,𝑝 (𝑓; [𝑎, 𝑏]) . ((𝑘 − 1)!)𝑝

(3.82)

Conversely, suppose now that 𝑓 : [𝑎, 𝑏] → ℝ belongs to 𝐴𝐶𝑘−1 ([𝑎, 𝑏]) and 𝑓(𝑘) ∈ 𝐿 𝑝 ([𝑎, 𝑏]). Let 𝑃 be a partition of [𝑎, 𝑏] of the form (3.81). Since 𝑓(𝑘−1) ∈ 𝐴𝐶([𝑎, 𝑏]) and 𝑓(𝑘) ∈ 𝐿 𝑝 ([𝑎, 𝑏]), by Exercise 2.49, we have 𝑚

∑

|𝑓[𝑡𝑗,𝑘+1 , . . . , 𝑡𝑗,2𝑘 ] − 𝑓[𝑡𝑗,1 , . . . , 𝑡𝑗,𝑘 ]|𝑝 |𝑡𝑗,2𝑘 − 𝑡𝑗,1 |𝑝−1

𝑗=1 𝑚

=∑ 𝑗=1

|𝑓(𝑘−1) (𝜏𝑗,2𝑘 ) − 𝑓(𝑘−1) (𝜏𝑗−1,𝑘 )|𝑝 ((𝑘 − 1)!)𝑝 |𝑡𝑗,2𝑘 − 𝑡𝑗,1 |𝑝−1

𝑚 1 1 = ∑ 𝑝 ((𝑘 − 1)!) 𝑗=1 |𝑡𝑗,2𝑘 − 𝑡𝑗,1 |𝑝−1

󵄨󵄨 𝜏𝑗 ,2𝑘 󵄨󵄨𝑝 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 (𝑘) 󵄨󵄨 ∫ 𝑓 (𝜎) 𝑑𝜎󵄨󵄨󵄨 , 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨𝜏𝑗−1,𝑘 󵄨󵄨 󵄨

3.6 Higher order Riesz-type variations

|

247

where 𝜏𝑗,2𝑘 belongs to the convex hull of {𝑡𝑗,𝑘+1 , . . . , 𝑡𝑗,2𝑘 } and 𝜏𝑗−1,𝑘 belongs to the convex hull of {𝑡𝑗,1 , . . . , 𝑡𝑗,𝑘 }. By Hölder’s inequality (0.108), we then get the estimates 𝑚

1 ∑ |𝑡 − 𝑡𝑗,1 |𝑝−1 𝑗=1 𝑗,2𝑘 𝑚

≤∑

󵄨󵄨 𝜏𝑗,2𝑘 󵄨󵄨󵄨𝑝 󵄨󵄨 󵄨󵄨 󵄨󵄨 (𝑘) 󵄨 󵄨󵄨󵄨 ∫ 𝑓 (𝑡) 𝑑𝑡󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨𝜏𝑗−1,𝑘

|𝜏𝑗,2𝑘 − 𝜏𝑗−1,𝑘 |𝑝−1 |𝑡𝑗,2𝑘 − 𝑡𝑗,1 |𝑝−1

𝑗=1

𝜏𝑗,2𝑘

∫ |𝑓(𝑘) (𝑡)|𝑝 𝑑𝑡 𝜏𝑗−1,𝑘

𝜏𝑗,2𝑘

𝑚

(𝑘)

𝑚

𝑝

𝑡𝑗,2𝑘

𝑏

≤ ∑ ∫ |𝑓 (𝑡)| 𝑑𝑡 ≤ ∑ ∫ |𝑓(𝑘) (𝑡)|𝑝 𝑑𝑡 = ∫ |𝑓(𝑘) (𝑡)|𝑝 𝑑𝑡 . 𝑗=1 𝜏

𝑗=1 𝑡

𝑗−1,𝑘

𝑎

𝑗,1

This implies that 𝑓 ∈ 𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]) and 𝑏

∫ 𝑎

|𝑓(𝑘) (𝑡)|𝑝 𝑑𝑡 ≥ Var𝑅𝑘,𝑝 (𝑓; [𝑎, 𝑏]) . ((𝑘 − 1)!)𝑝

(3.83)

Combining the estimates (3.82) and (3.83), we conclude that (3.80) is true, and so we are done. In the following Table 3.4 which is similar to Table 2.9, we recall the definition of higher order variations in Riesz’s sense, together with corresponding integral repre­ sentations. Table 3.4. Higher order variations and Riesz spaces 𝑅𝐵𝑉𝑘,𝑝 . 𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏])

𝑝=1

1 0 : Var𝑅𝑘,𝜙 (𝑓/𝜆; [𝑎, 𝑏]) ≤ 1} .

(3.86)

𝑗=0

As for (𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]), ‖ ⋅ ‖𝑅𝐵𝑉𝑘,𝑝 ), one may show that (𝑅𝐵𝑉𝑘,𝜙 ([𝑎, 𝑏]), ‖ ⋅ ‖𝑅𝐵𝑉𝑘,𝜙 ) is a Banach space. Let us take a closer look at the two particular cases 𝑘 = 1 and 𝑘 = 2. For 𝑘 = 1, the definitions (3.84)–(3.86) become 𝑚

Var𝑅1,𝜙 (𝑓, 𝑃; [𝑎, 𝑏]) = ∑ 𝜙 ( 𝑗=1

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| 𝑡𝑗 − 𝑡𝑗−1

) |𝑡𝑗 − 𝑡𝑗−1 | ,

Var𝑅1,𝜙 (𝑓; [𝑎, 𝑏]) := sup {Var𝑅1,𝜙 (𝑓, 𝑃; [𝑎, 𝑏]) : 𝑃 ∈ P([𝑎, 𝑏])} , 𝑅𝐵𝑉1,𝜙 ([𝑎, 𝑏]) = {𝑓 ∈ 𝐵([𝑎, 𝑏]) :

Var𝑅1,𝜙 (𝑐𝑓; [𝑎, 𝑏])

(3.87) (3.88)

< ∞ for some 𝑐 > 0} ,

and ‖𝑓‖𝑅𝐵𝑉1,𝜙 = |𝑓(𝑎)| + inf {𝜆 > 0 : Var𝑅1,𝜙 (𝑓/𝜆; [𝑎, 𝑏]) ≤ 1} ,

(3.89)

respectively, which gives Medvedev’s definition of the space 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]). For 𝑘 = 2 we get 𝑚−1 |𝑓[𝑡𝑗 , 𝑡𝑗+1 ] − 𝑓[𝑡𝑗−1 , 𝑡𝑗 ]| Var𝑅2,𝜙 (𝑓, 𝑃; [𝑎, 𝑏]) = ∑ 𝜙 ( ) |𝑡𝑗 − 𝑡𝑗−1 | , (3.90) 𝑡𝑗 − 𝑡𝑗−1 𝑗=1 Var𝑅2,𝜙 (𝑓; [𝑎, 𝑏]) = sup {Var𝑅2,𝜙 (𝑓, 𝑃; [𝑎, 𝑏]) : 𝑃 ∈ P([𝑎, 𝑏])} ,

and

(3.91)

𝑅𝐵𝑉2,𝜙 ([𝑎, 𝑏]) = {𝑓 ∈ 𝐵([𝑎, 𝑏]) : Var𝑅2,𝜙 (𝑐𝑓; [𝑎, 𝑏]) < ∞ for some 𝑐 > 0} ,

(3.92)

‖𝑓‖𝑅𝐵𝑉2,𝜙 = |𝑓(𝑎)| + |𝑓󸀠 (𝑎)| + inf {𝜆 > 0 : Var𝑅2,𝜙 (𝑓/𝜆; [𝑎, 𝑏]) ≤ 1} ,

(3.93)

respectively. The space (𝑅𝐵𝑉2,𝜙 ([𝑎, 𝑏]), ‖ ⋅ ‖𝑅𝐵𝑉2,𝜙 ) in this form has been introduced and studied in [218], where also an analogue to Theorem 3.39 is proved.

3.7 Comments on Chapter 3 | 249

3.7 Comments on Chapter 3 The basic facts about the interconnections between absolutely continuous functions and functions of bounded variation may be found in the books [76, 156, 182, 188, 238, 266, 311]. More sophisticated (and in part, surprising) properties of absolutely continu­ ous functions are given in the survey papers [44, 45]. As Section 3.4 shows, absolutely continuous functions of one variable are intimately related to the concept of curve length. The analogue in higher dimensions is surface area; a beautiful discussion can be found in Goffman’s survey [126]. Proposition 3.2 shows that the requirement |𝑔(𝑥)| > 0 for 𝑔 ∈ 𝐴𝐶 suffices to en­ sure that 1/𝑔 ∈ 𝐴𝐶, while from Exercises 1.1 and 1.2, it follows that 𝑔 ∈ 𝐵𝑉 has to be bounded away from zero to ensure¹⁸ that 1/𝑔 ∈ 𝐵𝑉. The construction and properties of the Cantor set (and similar sets of positive measure) can be found in many standard textbooks on Real Analysis, e.g. [118], the important Vitali–Banach–Zaretskij theorem (under this or another name), for example, in [156, 182, 238]. The idea to extend the Cantor function (3.6) from the Cantor set (3.3) to the whole interval [0, 1], preserving its monotonicity, is based on the following general principle: if 𝐶 ⊂ [𝑎, 𝑏] is a subset with 𝑎, 𝑏 ∈ 𝐶 and 𝑓 : 𝐶 → ℝ is increasing, then 𝑓 extends to an increasing function 𝑓 ̂ : [𝑎, 𝑏] → ℝ. In fact, defining ̂ := sup {𝑓(𝑡) : 𝑡 ∈ 𝐶 ∩ [𝑎, 𝑥]} , 𝑓(𝑥)

(3.94)

̂ = 𝑓(𝑥) for 𝑥 ∈ 𝐶. one easily verifies that 𝑓 ̂ is increasing and 𝑓(𝑥) Our discussion of the class 𝐴𝐶𝑝 ([𝑎, 𝑏]) is taken from [183]. Table 3.2 shows that 𝐴𝐶𝑝 ⊆ 𝑊𝐵𝑉𝑝 for 𝑝 > 1 (even ⊆ 𝐶 ∩ 𝑊𝐵𝑉𝑝 since all functions in 𝐴𝐶𝑝 are continuous). More precisely, in [185] the chain of inclusions (3.14) is proved. Roughly speaking, this means that the difference between the spaces 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) and 𝐴𝐶𝑝 ([𝑎, 𝑏]) is “not very large.” As pointed out before, however, all inclusions in (3.14) are strict. We remark that Bruneau [65] defines 𝑝-absolutely continuous functions 𝑓 : [𝑎, 𝑏] → ℝ by requiring that for each 𝜀 > 0, there exists some 𝛿 > 0 such that 𝑛

∑ Var𝑊 𝑝 (𝑓; [𝑎𝑘 , 𝑏𝑘 ]) ≤ 𝜀

𝑘=1

for any 𝑆 ∈ 𝛴([𝑎, 𝑏]) with 𝛩(𝑆) ≤ 𝛿. As Proposition 3.14 shows, this coincides with Love’s definition (our Definition 3.12) from [183]. Interestingly, such functions may be characterized differently: as the following theorem [65] shows, 𝐴𝐶𝑝 may be regarded as closure of 𝐿𝑖𝑝1/𝑝 in the 𝑊𝐵𝑉𝑝 -norm (1.65). Theorem 3.41. A function 𝑓 : [𝑎, 𝑏] → ℝ is 𝑝-absolutely continuous if and only if there exists a sequence (𝑓𝑛 )𝑛 in 𝐿𝑖𝑝1/𝑝 ([𝑎, 𝑏]) such that ‖𝑓𝑛 − 𝑓‖𝑊𝐵𝑉𝑝 → 0 as 𝑛 → ∞.

18 This is of course not surprising since 𝐵𝑉 function are bounded, but not necessarily continuous.

250 | 3 Absolutely continuous functions In [2], the author introduces and studies the four metrics 𝑑1 (𝑓, 𝑔) := ‖𝑓 − 𝑔‖𝐿 1 + | Var(𝑓; [𝑎, 𝑏]) − Var(𝑔; [𝑎, 𝑏])| 𝑏

(3.95)

= ∫ |𝑓(𝑡) − 𝑔(𝑡)| 𝑑𝑡 + | Var(𝑓; [𝑎, 𝑏]) − Var(𝑔; [𝑎, 𝑏])| , 𝑎

𝑑2 (𝑓, 𝑔) := ‖𝑓 − 𝑔‖𝐿 1 + |𝐿(𝛤(𝑓); [𝑎, 𝑏]) − 𝐿(𝛤(𝑔); [𝑎, 𝑏])| 𝑏

(3.96)

= ∫ |𝑓(𝑡) − 𝑔(𝑡)| 𝑑𝑡 + |𝐿(𝛤(𝑓); [𝑎, 𝑏]) − 𝐿(𝛤(𝑔); [𝑎, 𝑏])| , 𝑎

𝑑3 (𝑓, 𝑔) := ‖𝑓 − 𝑔‖∞ + | Var(𝑓; [𝑎, 𝑏]) − Var(𝑔; [𝑎, 𝑏])| = sup |𝑓(𝑥) − 𝑔(𝑥)| + | Var(𝑓; [𝑎, 𝑏]) − Var(𝑔; [𝑎, 𝑏])| ,

(3.97)

𝑎≤𝑥≤𝑏

and 𝑑4 (𝑓, 𝑔) := ‖𝑓 − 𝑔‖∞ + |𝐿(𝛤(𝑓); [𝑎, 𝑏]) − 𝐿(𝛤(𝑔); [𝑎, 𝑏])| = sup |𝑓(𝑥) − 𝑔(𝑥)| + |𝐿(𝛤(𝑓); [𝑎, 𝑏]) − 𝐿(𝛤(𝑔); [𝑎, 𝑏])| ,

(3.98)

𝑎≤𝑥≤𝑏

on the space 𝐵𝑉([𝑎, 𝑏]). Clearly, 𝑑1 (𝑓, 𝑔) ≤ 𝑑3 (𝑓, 𝑔) and 𝑑2 (𝑓, 𝑔) ≤ 𝑑4 (𝑓, 𝑔). Moreover, from (3.59), it follows that 𝑑1 (𝑓, 𝑔) ≤ 𝑑2 (𝑓, 𝑔) and 𝑑3 (𝑓, 𝑔) ≤ 𝑑4 (𝑓, 𝑔). We may summa­ rize these inequalities in the form 𝑑1 (𝑓, 𝑔) ≤ min {𝑑2 (𝑓, 𝑔), 𝑑3 (𝑓, 𝑔)} ≤ max {𝑑2 (𝑓, 𝑔), 𝑑3 (𝑓, 𝑔)} ≤ 𝑑4 (𝑓, 𝑔) .

(3.99)

As is shown in [2], the topological properties of 𝐵𝑉 are quite different for each choice of these metrics. For example, equipped with either the metric (3.97) or (3.98), the spaces 𝐵𝑉([𝑎, 𝑏]) and 𝐴𝐶([𝑎, 𝑏]) are not complete and not even locally compact. On the other hand, in 𝐵𝑉([𝑎, 𝑏]) with the metric (3.95), every closed ball is compact.¹⁹ The following Luzin type result is mentioned in the papers [128, 129]: Theorem 3.42 (Goffman–Liu). A function 𝑓 : [𝑎, 𝑏] → ℝ is absolutely continuous if and only if for each 𝜀 > 0, we can find a continuously differentiable function 𝑓𝜀 : [𝑎, 𝑏] → ℝ such that ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 < 𝜀, ∫ |𝑓𝜀󸀠 (𝑡)| 𝑑𝑡 < 𝜀 , 𝜆(𝛥(𝑓, 𝑓𝜀 )) < 𝜀, 𝛥(𝑓,𝑓𝜀 )

𝛥(𝑓,𝑓𝜀 )

where 𝛥(𝑓, 𝑔) is given by (0.47). A comparison of Theorem 3.42 and Theorem 0.34 shows that the pair “measurable vs. continuous” in the Luzin theorem corresponds to the pair “𝐴𝐶 vs. 𝐶1 ” in the Goff­

19 This is a consequence of Helly’s selection theorem (Theorem 1.11) and Lebesgue’s convergence the­ orem (Theorem 0.4). Loosely speaking, it means that (𝐵𝑉, 𝑑1 ) behaves rather like a finite dimensional space.

3.7 Comments on Chapter 3 |

251

man–Liu theorem; moreover, for the latter, we obviously have to take into account the 𝐿 1 -norm of the derivatives. Theorem 3.42 suggests to equip the set 𝐴𝐶([𝑎, 𝑏]) with the metric 𝑑(𝑓, 𝑔) := 𝜆(𝛥(𝑓, 𝑔)) + ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 + ∫ |𝑔󸀠 (𝑡)| 𝑑𝑡 . 𝛥(𝑓,𝑔)

(3.100)

𝛥(𝑓,𝑔)

As far as we know, the metric space (𝐴𝐶, 𝑑) has not been studied yet in the litera­ ture.²⁰ The results discussed in Section 3.3 may also be found in [76] or [156]. In the inter­ esting survey article [309], the author cites the following proposition which he found “buried as an innocent problem” in Natanson’s book [238]: Proposition 3.43. Let 𝑓 : [𝑎, 𝑏] → ℝ be differentiable with bounded derivative on a set 𝑀 ⊆ [𝑎, 𝑏]. Then 𝜆∗ (𝑓(𝑀)) ≤ 𝜆∗ (𝑀) sup |𝑓󸀠 (𝑥)| , (3.101) 𝑥∈𝑀

∗

where 𝜆 (𝑀) denotes the outer Lebesgue measure of 𝑀. From Proposition 3.43, several results (among them, many of those presented in Sec­ tion 3.3) may be derived rather easily. For example, if 𝐷𝑒𝑟0 (𝑓) := {𝑥 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑓󸀠 (𝑥) = 0}

(3.102)

denotes the set of critical points of 𝑓 : [𝑎, 𝑏] → ℝ, (3.101) implies that the set of critical values 𝑓(𝐷𝑒𝑟0 (𝑓)) = {𝑓(𝑥) : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑓󸀠 (𝑥) = 0} (3.103) of 𝑓 is always a nullset²¹ and therefore “small.” Of course, constant functions show that the set (3.102) may be very large. It is instructive to compare the sets (3.102) and (3.103) with the set of extremal points 𝐸𝑥𝑡(𝑓) := {𝑥 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑥 is a local extremum for 𝑓}

(3.104)

and the set of extremal values 𝑓(𝐸𝑥𝑡(𝑓)) = {𝑓(𝑥) : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑥 is a local extremum for 𝑓}

(3.105)

of 𝑓. Again, constant functions show that the set (3.104) may be very large, while the set (3.105) may be a singleton. Even more interesting is the characteristic function 𝑓 of [𝑎, 𝑏]∩ℚ (Dirichlet function) which satisfies 𝐸𝑥𝑡(𝑓) = [𝑎, 𝑏] and 𝑓(𝐸𝑥𝑡(𝑓)) = {0, 1}. This function has a local extremum at every point, but is not constant on any subinterval.²²

20 In [129], it is mentioned without proof that 𝐴𝐶 with the metric (3.100) is complete. 21 This result is known as Sard’s lemma in the literature. We remark that this result is rather easy to prove for 𝐶1 -functions. 22 One may show that such a function must be discontinuous at every point, see Exercise 3.59.

252 | 3 Absolutely continuous functions In every calculus course, it is taught that 𝐸𝑥𝑡(𝑓) ⊆ 𝐷𝑒𝑟0 (𝑓) for differentiable func­ tions 𝑓. Interestingly, the set (3.105) is even “smaller” than just a nullset, as the set (3.103). In fact, let 𝑦̂ = 𝑓(𝑥)̂ ∈ 𝑓(𝐸𝑥𝑡(𝑓)) be such that 𝑓 has a local maximum at ̂ 𝑏(𝑦)] ̂ with rational endpoints such that 𝑥,̂ say, and choose an interval 𝐼(𝑦)̂ = [𝑎(𝑦), ̂ Since the map 𝑦̂ 󳨃→ 𝐼(𝑦)̂ is injective, and the system of 𝑓(𝑥) ≤ 𝑓(𝑥)̂ for 𝑎(𝑥)̂ ≤ 𝑥 ≤ 𝑏(𝑥). all such intervals 𝐼(𝑦)̂ is countable, we thus have proved²³ the following Proposition 3.44. For arbitrary functions 𝑓 : [𝑎, 𝑏] → ℝ, the set (3.105) of extremal values of 𝑓 is at most countable. A certain converse of Proposition 3.44 is true as well, see Exercise 3.58. The following example [12] illustrates the fact that although both sets (3.103) and (3.105) are “small,” the gap between them may be considerable: Example 3.45. Let 𝐶 be the Cantor set (3.3), and let 𝑔 : [0, 1] → ℝ and 𝑓 : [0, 1] → ℝ be defined by 𝑥

𝑔(𝑡) := dist(𝑡, 𝐶) = min {𝑡 − 𝑐 : 𝑐 ∈ 𝐶},

𝑓(𝑥) := ∫ 𝑔(𝑡) 𝑑𝑡 . 0

Then 𝑔 ∈ 𝐿𝑖𝑝([0, 1]) with 𝑔(𝑡) = 0 on 𝐶 and 𝑔(𝑡) > 0 on [0, 1] \ 𝐶. Consequently, 𝑓 ∈ 𝐶1 ([0, 1]) is strictly increasing, and so 𝑓(𝐸𝑥𝑡(𝑓)) = {0, 𝑓(1)}. On the other hand, 𝑓󸀠 (𝑥) = 0 on 𝐶, which means that 𝐷𝑒𝑟0 (𝑓) contains the uncountable set 𝐶. However, the injectivity of 𝑓 implies that then 𝑓(𝐷𝑒𝑟0 (𝑓)) contains the uncountable set 𝑓(𝐶), and so is uncountable (although a nullset) itself. ♥ As we have seen in Example 3.15, there exist simple functions 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) which are everywhere differentiable on [𝑎, 𝑏] and have the property that 𝑓󸀠 is not Lebesgue integrable. Of course, as Theorem 3.18 shows, such a function cannot be absolutely continuous. A more dramatic refinement of this is an example, due to Katznelson and Stromberg²⁴ [157] of a function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) with the following properties: – 𝑓 is everywhere differentiable on [𝑎, 𝑏] with ‖𝑓󸀠 ‖∞ ≤ 1; – both sets {𝑥 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑓󸀠 (𝑥) > 0} and {𝑥 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑓󸀠 (𝑥) < 0} are dense in [𝑎, 𝑏], so 𝑓 is not monotone on any interval; – the set {𝑥 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑓󸀠 (𝑥) = 0} is also dense in [𝑎, 𝑏]; – 𝑓󸀠 is not Riemann integrable over any interval [𝑐, 𝑑] ⊆ [𝑎, 𝑏].

23 The reader will have noticed that the proof of this result is similar to the proof of the countability of the set of discontinuities of a monotone function. 24 An explicit construction of this function is given in Example 13.2 of the book [268]. A more el­ ementary example of a function 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) which is differentiable everywhere (with a bounded derivative) and such that 𝑓󸀠 vanishes on a dense subset of [𝑎, 𝑏] may be found in Example 13.3 of [268].

3.7 Comments on Chapter 3 | 253

Jordan [153] has not only introduced the concept of bounded variation, but also stud­ ied functions of bounded variation. The connection between functions of bounded variation and rectifiable curves in the plane (Proposition 3.28 (c)) is also due to Jordan [154]. Theorem 3.25 which is taken from the paper [127] admits the following geomet­ ric interpretation. The family {𝑔ℎ : 0 < ℎ < 1} defined in (3.44) converges, as ℎ → 0+, in 𝐴𝐶([0, 1]) to the boundary of a ball in 𝐴𝐶([0, 1]) centered at the origin if and only if 𝑔 is (equivalent to) a function of bounded variation, and it converges to a point in 𝐴𝐶([0, 1]) if and only if 𝑔 is (equivalent to) an absolutely continuous function. The Riesz theorem (Theorem 3.34) and its generalization by Medvedev (Theo­ rem 3.36) are of utmost importance in functional analysis and in the theory of dif­ ferential equations. Roughly speaking, passing from the Riesz space 𝑅𝐵𝑉𝑝 to the Medvedev space 𝑅𝐵𝑉𝜙 plays a similar role in applications to differential equations as passing from the Lebesgue space 𝐿 𝑝 to the Orlicz space 𝐿 𝜙 in applications to integral equations. In [215], it is shown that Theorem 3.34 also holds for 𝑓 replaced by 𝑓∨ (𝑥) := sup {

𝑓(𝑥 + ℎ) − 𝑓(𝑥 − ℎ) : 0 < ℎ ≤ min{𝑏 − 𝑥, 𝑥 − 𝑎}} 2ℎ

for 𝑎 < 𝑥 < 𝑏. We remark that the function 𝑓∨ occurs in the theory of maximal opera­ tors in Hardy’s sense. More on integral representations of functions of bounded Riesz variation may be found in [283]. In view of their importance, let us summarize the main results of Section 3.3 in the following way, where we also consider the variation function (1.13). Theorem 3.46. The following four assertions on a function 𝑓 : [𝑎, 𝑏] → ℝ are equiva­ lent. (a) 𝑓 is absolutely continuous; (b) 𝑓󸀠 exists a.e., 𝑓󸀠 ∈ 𝐿 1 ([𝑎, 𝑏]), and (3.31) holds; (c) 𝑓󸀠 exists a.e., 𝑓󸀠 ∈ 𝐿 1 ([𝑎, 𝑏]), and (3.36) holds; (d) 𝑉𝑓 is absolutely continuous. The implication (a) ⇒ (b) has been proved in Theorem 3.18, while the implication (b) ⇒ (c) has been discussed after the proof of Theorem 3.19. The implication (c) ⇒ (d) follows from (3.33), while the implication (d) ⇒ (a) is a consequence of the estimates |𝑓(𝑥) − 𝑓(𝑦)| ≤ Var(𝑓; [𝑥, 𝑦]) ≤ 𝑉𝑓 (𝑦) − 𝑉𝑓 (𝑥) which follow from (1.8) and (1.10). The results proved in Section 3.3 imply some interesting decompositions. Writing as before 𝐴𝐶𝑜 ([𝑎, 𝑏]) for the space of all 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) satisfying 𝑓(𝑎) = 0, we have the trivial decomposition 𝐴𝐶([𝑎, 𝑏]) = ℝ ⊕ 𝐴𝐶𝑜 ([𝑎, 𝑏]) (3.106)

254 | 3 Absolutely continuous functions induced by the linear surjective isometry 𝑓 󳨃→ 𝑓(𝑎) + (𝑓 − 𝑓(𝑎)). Defining the integral operator 𝐽 by 𝑥

𝐽𝑓(𝑥) := ∫ 𝑓(𝑡) 𝑑𝑡 ,

(3.107)

𝑎

Proposition 3.4 and Theorem 3.18 show that 𝐽 is a linear surjective isometry be­ tween 𝐿 1 ([𝑎, 𝑏]) and 𝐴𝐶𝑜 ([𝑎, 𝑏]) with inverse 𝐽−1 𝑔 = 𝐷𝑔 = 𝑔󸀠 . Thus, we may replace (3.106) by the more interesting decomposition 𝐴𝐶([𝑎, 𝑏]) = ℝ ⊕ 𝐿 1 ([𝑎, 𝑏])

(3.108)

induced by the linear surjective isometry 𝑓 󳨃→ 𝑓(𝑎) + 𝑓󸀠 . As a pleasant by-product, we get a very simple proof of the completeness of the space 𝐴𝐶: since 𝐿 1 is complete and 𝐽 is an isometry, 𝐴𝐶𝑜 = 𝐽(𝐿 1 ) is complete as well, and so 𝐴𝐶 is complete, by (3.106). The higher order Riesz–Young–Medvedev spaces which we discussed in Sec­ tion 3.6 have been studied in [226, 228, 229] and elsewhere. In [228], it is shown that 𝑅𝐵𝑉𝑘,𝜙 ⊆ 𝑊𝐵𝑉𝑘,1 , with equality in case 𝜙 ∈ ̸ ∞1 , see (2.16). The important equality 𝑏

Var𝑅2,𝑝 (𝑓; [𝑎, 𝑏])

= ∫ |𝑓󸀠󸀠 (𝑥)|𝑝 𝑑𝑥

(3.109)

𝑎

which holds in case 𝑓󸀠 ∈ 𝐴𝐶([𝑎, 𝑏]) (Table 3.4) and is analogous to (3.69) for 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) may be found in [220]. As far as we know, a rigorous proof of the general formula (3.80) appears in case 𝑓(𝑘−1) ∈ 𝐴𝐶([𝑎, 𝑏]) for the first time in [229]. In the paper [176], the authors also prove a parallel result to Theorem 3.39. To formulate this result, we again need the Orlicz class L𝜙 ([𝑎, 𝑏]) which we introduced in Definition 0.16. Theorem 3.47. Let 𝜙 be a Young function which satisfies condition ∞1 , see (2.16), and let 𝑘 ∈ ℕ. Then a function 𝑓 belongs to 𝑅𝐵𝑉𝑘,𝜙 ([𝑎, 𝑏]) if and only if 𝑓 ∈ 𝐴𝐶𝑘−1 ([𝑎, 𝑏]) and 𝑓(𝑘) ∈ L𝜙 ([𝑎, 𝑏]). Moreover, in this case, the equality 𝑏

Var𝑅𝑘,𝜙 (𝑓; [𝑎, 𝑏])

= ∫𝜙( 𝑎

|𝑓(𝑘) (𝑡)| ) 𝑑𝑡 (𝑘 − 1)!

(3.110)

holds, where Var𝑅𝑘,𝜙 (𝑓; [𝑎, 𝑏]) denotes the (𝑘, 𝜙)-variation (3.85) of 𝑓 in Riesz’s sense. Relations between functions of (generalized) bounded variation and absolutely con­ tinuous functions have been also studied in a more general setting. For instance, [159] shows that a function 𝑓 ∈ Λ𝐵𝑉𝜙 ([𝑎, 𝑏]) (Definition 2.84) is absolutely continuous if and only if 𝑓󸀠 exists and satisfies (3.18). Since Λ𝐵𝑉𝜙 contains the spaces Λ𝐵𝑉 and 𝑊𝐵𝑉𝜙 , this is a far reaching generalization of Theorem 3.18. The Riesz–Medvedev theorem has a natural analogue for functions of two vari­ ables. However, formulating this analogue requires some preparation.

3.7 Comments on Chapter 3 | 255

We start with the definition of the corresponding variation, imitating what we have done in Definition 1.42. Consider a Young function 𝜙 : [0, ∞) → [0, ∞), a func­ tion, 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ, and partitions 𝑃 = {𝑠0 , 𝑠1 , . . . , 𝑠𝑚 } ∈ P([𝑎, 𝑏]) and 𝑄 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑛 } ∈ P([𝑐, 𝑑]). We define three variations for 𝑓 with respect to 𝑃 and 𝑄 by 𝑚

Var𝑅𝜙 (𝑓(⋅, 𝑐), 𝑃; [𝑎, 𝑏]) := ∑ 𝜙 (

|𝑓(𝑠𝑖 , 𝑐) − 𝑓(𝑠𝑖−1 , 𝑐)| ) (𝑠𝑖 − 𝑠𝑖−1 ) , 𝑠𝑖 − 𝑠𝑖−1

Var𝑅𝜙 (𝑓(𝑎, ⋅), 𝑄; [𝑐, 𝑑]) := ∑ 𝜙 (

|𝑓(𝑎, 𝑡𝑗 ) − 𝑓(𝑎, 𝑡𝑗−1 )|

𝑖=1 𝑛

𝑗=1

𝑡𝑗 − 𝑡𝑗−1

) (𝑡𝑗 − 𝑡𝑗−1 ) ,

(3.111) (3.112)

and V𝑅2,𝜙 (𝑓, 𝑃 × 𝑄; [𝑎, 𝑏] × [𝑐, 𝑑]) 𝑚

𝑛

:= ∑ ∑ 𝜙 ( 𝑖=1 𝑗=1

|𝛥 𝑖,𝑗 𝑓| (𝑠𝑖 − 𝑠𝑖−1 )(𝑡𝑗 − 𝑡𝑗−1 )

) (𝑠𝑖 − 𝑠𝑖−1 )(𝑡𝑗 − 𝑡𝑗−1 ) ,

(3.113)

where we have used the shortcut 𝛥 𝑖,𝑗 𝑓 := 𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑓(𝑠𝑖−1 , 𝑡𝑗 ) − 𝑓(𝑠𝑖 , 𝑡𝑗−1 ) + 𝑓(𝑠𝑖−1 , 𝑡𝑗−1 ) .

(3.114)

Of course, in case 𝜙(𝑡) = 𝑡, these variations reduce to the three variations (1.76), (1.77), and (1.78). As in Section 1.4, we put Var𝑅𝜙 (𝑓(⋅, 𝑐); [𝑎, 𝑏]) := sup {Var𝑅𝜙 (𝑓(⋅, 𝑐), 𝑃; [𝑎, 𝑏]) : 𝑃 ∈ P([𝑎, 𝑏])} ,

(3.115)

Var𝑅𝜙 (𝑓(𝑎, ⋅); [𝑐, 𝑑]) := sup {Var𝑅𝜙 (𝑓(𝑎, ⋅), 𝑄; [𝑐, 𝑑]) : 𝑄 ∈ P([𝑐, 𝑑])} ,

(3.116)

and V𝑅2,𝜙 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) := sup {V𝑅2,𝜙 (𝑓, 𝑃 × 𝑄; [𝑎, 𝑏] × [𝑐, 𝑑]) : 𝑃 ∈ P([𝑎, 𝑏]), 𝑄 ∈ P([𝑐, 𝑑])} ,

(3.117)

where all suprema are taken over the indicated partitions. Definition 3.48. With the above notation, we call the (possibly infinite) number Var𝑅𝜙 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) := Var𝑅𝜙 (𝑓(⋅, 𝑐); [𝑎, 𝑏]) + Var𝑅𝜙 (𝑓(𝑎, ⋅); [𝑐, 𝑑]) + V𝑅2,𝜙 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑])

(3.118)

the total variation of 𝑓 on [𝑎, 𝑏] × [𝑐, 𝑑] in the sense of Riesz–Medvedev. In case Var𝑅𝜙 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) < ∞, we say that 𝑓 has bounded Riesz–Medvedev variation on [𝑎, 𝑏] × [𝑐, 𝑑] and write 𝑓 ∈ 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏] × [𝑐, 𝑑]). ◼ By what we have observed before, we get 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏] × [𝑐, 𝑑]) = 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) for the special choice 𝜙(𝑡) = 𝑡. We point out, however, that not all properties of the onedimensional case carry over to this space. For example, in Proposition 2.52 we have proved the inclusions 𝐿𝑖𝑝([𝑎, 𝑏]) ⊆ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) ⊆ 𝐴𝐶([𝑎, 𝑏])

(3.119)

256 | 3 Absolutely continuous functions for 𝑝 > 1. It is a striking fact that the first inclusion in (3.119) is not true in the two-di­ mensional case, even for the simplest Young function 𝜙(𝑡) = 𝑡𝑝 , as the following ex­ ample from [33] shows: Example 3.49. On the unit square [0, 1]×[0, 1], consider for 𝑛 = 2, 3, 4, . . . the partitions 𝑃𝑛 := {𝑠0 , 𝑠1 , 𝑠2 , . . . , 𝑠2𝑛−1 },

𝑄𝑛 := {0, 𝑡1 , 𝑡2 , . . . , 𝑡2𝑛−1 } ,

where 𝑠0 = 𝑡0 := 0, ...,

𝑠2𝑛−3 = 𝑡2𝑛−3 :=

𝑠1 = 𝑡1 := 1 , 2

1 , 𝑛

𝑠2 = 𝑡2 :=

𝑠2𝑛−2 = 𝑡2𝑛−2 :=

3 , 4

2𝑛 − 1 , 2𝑛(𝑛 − 1)

𝑠3 = 𝑡3 :=

1 , 𝑛−1

...

𝑠2𝑛−1 = 𝑡2𝑛−1 := 1 .

Thus, 𝑃𝑛 × 𝑄𝑛 is an equidistant lattice of constant mesh size 1/𝑛(𝑛 − 1). We define a function 𝑓 : [0, 1] × [0, 1] → ℝ by 0 { { { 1 𝑓(𝑥, 𝑦) := { 𝑘(𝑘−1) { { {linear

for 𝑥 = 𝑠2𝑖−1 or 𝑦 = 𝑡2𝑗−1 , for 𝑥 = 𝑦 =

2𝑘−1 2𝑘(𝑘−1)

,

otherwise .

Geometrically, the graph of 𝑓 over the rectangle 1 1 1 1 ]×[ , ] 𝑅𝑖,𝑗 := [ , 𝑖 𝑖−1 𝑗 𝑗−1 is the surface of a pyramid with vertex situated at the midpoint 𝑃𝑖,𝑗 := (

2𝑗 − 1 1 2𝑖 − 1 , , ). 2𝑖(𝑖 − 1) 2𝑗(𝑗 − 1) 𝑖(𝑖 − 1)

Evaluating the slope of any such pyramid, it is not hard to see that ̃ 2≤ |𝑓(𝑥, 𝑦) − 𝑓(𝑥,̃ 𝑦)|

1 (|𝑥 − 𝑥|̃ 2 + |𝑦 − 𝑦|̃ 2 )‖ , 4

(3.120)

which shows that 𝑓 is Lipschitz continuous in the Euclidean norm on [0, 1] × [0, 1]. On the other hand, even for the simple choice 𝜙(𝑡) = 𝑡2 , the variation (3.113) of 𝑓 is not finite, and so 𝑓 ∈ ̸ 𝑅𝐵𝑉𝜙 ([0, 1] × [0, 1]). To see this, we show that V𝑅2,𝜙 (𝑓, 𝑃𝑛 × 𝑄𝑛 ; [0, 1] × [0, 1]) → ∞ (𝑛 → ∞) ,

(3.121)

3.7 Comments on Chapter 3 |

257

where 𝑃𝑛 and 𝑄𝑛 are as before. Using the shortcut (3.114) and taking into account that 𝜙(𝑡) = 𝑡2 , we get the chain of equalities V𝑅2,𝜙 (𝑓, 𝑃𝑛 × 𝑄𝑛 ; [0, 1] × [0, 1]) 2𝑛−1 2𝑛−1

= ∑ ∑ 𝜙( 𝑖=1 𝑗=1 2𝑛−1 2𝑛−1

= ∑ ∑ 𝜙( 𝑖=2 𝑗=2 2𝑛−1 2𝑛−1

= ∑ ∑ 𝑖=2 𝑗=2

|𝛥 𝑖,𝑗 𝑓| (𝑠𝑖 − 𝑠𝑖−1 )(𝑡𝑗 − 𝑡𝑗−1 ) 1 𝑖(𝑖−1) 1 1 𝑖(𝑖−1) 𝑗(𝑗−1)

)(

) (𝑠𝑖 − 𝑠𝑖−1 )(𝑡𝑗 − 𝑡𝑗−1 )

1 1 ) 𝑖(𝑖 − 1) 𝑗(𝑗 − 1)

2𝑛−1 2𝑛−1 𝑗2 (𝑗 − 1)2 𝑗(𝑗 − 1) = ∑ ∑ . 𝑖(𝑖 − 1)𝑗(𝑗 − 1) 𝑖(𝑖 − 1) 𝑖=2 𝑗=2

However, the second sum may be estimated for fixed 𝑖 ∈ {2, 3, . . . , 2𝑛 − 1} by 2𝑛−1

∑ 𝑗=2

𝑗(𝑗 − 1) (2𝑛 − 1)(2𝑛 − 2) + (2𝑛 − 2)(2𝑛 − 3) + . . . + 3 ⋅ 2 + 2 ⋅ 1 = 𝑖(𝑖 − 1) 𝑖(𝑖 − 1) ≥

(2𝑛 − 1) + (2𝑛 − 2) + . . . + 3 + 2 + 1 𝑛(2𝑛 − 1) = , 𝑖(𝑖 − 1) 𝑖(𝑖 − 1)

which implies 2𝑛−1

V𝑅2,𝜙 (𝑓, 𝑃𝑛 × 𝑄𝑛 ; [0, 1] × [0, 1]) ≥ ∑ 𝑖=2

𝑛(2𝑛 − 1) 𝑖(𝑖 − 1)

= 𝑛(2𝑛 − 1) (1 −

1 ) = 2𝑛(𝑛 − 1) . 2𝑛 − 1

This proves (3.121), and hence 𝑓 ∈ ̸ 𝑅𝐵𝑉𝜙 ([0, 1] × [0, 1]) as claimed.

♥

The reason for this counterexample is that the Lipschitz condition (3.120) is too weak to ensure that 𝑓 ∈ 𝑅𝐵𝑉𝜙 . In Proposition 3.2 of [33], it is shown that the combination of the three conditions |𝑓(𝑠𝑖 , 𝑡) − 𝑓(𝑠𝑖−1 , 𝑡)| ≤ 𝐿 1 (𝑡)|𝑠𝑖 − 𝑠𝑖−1 | (𝑐 ≤ 𝑡 ≤ 𝑑) , |𝑓(𝑠, 𝑡𝑗 ) − 𝑓(𝑠, 𝑡𝑗−1 )| ≤ 𝐿 2 (𝑠)|𝑡𝑗 − 𝑡𝑗−1 | (𝑎 ≤ 𝑠 ≤ 𝑏) , and |𝛥 𝑖𝑗 𝑓| ≤ 𝐿|(𝑠𝑖 − 𝑠𝑖−1 )(𝑡𝑗 − 𝑡𝑗−1 )| implies that 𝑚

Var𝑅𝜙 (𝑓(⋅, 𝑐); [𝑎, 𝑏]) ≤ 𝜙(𝐿 1 (𝑐)) ∑ |𝑠𝑖 − 𝑠𝑖−1 | , 𝑖=1 𝑛

Var𝑅𝜙 (𝑓(𝑎, ⋅); [𝑐, 𝑑]) ≤ 𝜙(𝐿 2 (𝑎)) ∑ |𝑡𝑗 − 𝑡𝑗−1 | , 𝑗=1

258 | 3 Absolutely continuous functions and

𝑚

𝑛

V𝑅2,𝜙 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) ≤ 𝜙(𝐿) ∑ ∑ |(𝑠𝑖 − 𝑠𝑖−1 )(𝑡𝑗 − 𝑡𝑗−1 )| . 𝑖=1 𝑗=1

Consequently, Var𝑅𝜙 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) ≤ 𝜙(𝐿 1 (𝑐))(𝑏 − 𝑎) + 𝜙(𝐿 2 (𝑎))(𝑑 − 𝑐) + 𝜙(𝐿)(𝑏 − 𝑎)(𝑑 − 𝑐) , and hence 𝑓 ∈ 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏] × [𝑐, 𝑑]). Now, we come to the definition of absolutely continuous functions of two vari­ ables. In [182], a definition is given which uses the concept of the “function of rect­ angles” and is rather technical. However, in [297], it is shown that this definition is equivalent to the following Definition 3.50. We call a function 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ absolutely continuous and write 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏] × [𝑐, 𝑑]) if 𝑓 admits a representation of the form 𝑦

𝑥

𝑥 𝑦

𝑓(𝑥, 𝑦) = 𝑓(𝑎, 𝑐) + ∫ 𝑔1 (𝑠) 𝑑𝑠 + ∫ 𝑔2 (𝑡) 𝑑𝑡 + ∫ ∫ 𝑔(𝑠, 𝑡) 𝑑𝑡 𝑑𝑠 , 𝑎

𝑐

𝑎 𝑐

where 𝑔1 ∈ 𝐿 1 ([𝑎, 𝑏]), 𝑔2 ∈ 𝐿 1 ([𝑐, 𝑑]), and 𝑔 ∈ 𝐿 1 ([𝑎, 𝑏] × [𝑐, 𝑑]).

◼

Definition 3.50 seems very natural since it is parallel to the fact that absolutely con­ tinuous functions of one variable are precisely primitives of 𝐿 1 -functions, see Propo­ sition 3.4 and Theorem 3.18. Another equivalent formulation is given in Exercise 3.62. With this notion of absolute continuity, the following theorem which is perfectly analogous to Theorem 3.36 and involves the partial derivatives 𝑓𝑥 , 𝑓𝑦 and 𝑓𝑥𝑦 of 𝑓, has been proved in [33]. Theorem 3.51. Let 𝜙 be a Young function which satisfies condition ∞1 , see (2.16). Then a function 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ belongs to 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏] × [𝑐, 𝑑]) if and only if 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏] × [𝑐, 𝑑]), 𝑓𝑥 (⋅, 𝑐) ∈ L𝜙 ([𝑎, 𝑏]), 𝑓𝑦 (𝑎, ⋅) ∈ L𝜙 ([𝑐, 𝑑]), and²⁵ 𝑓𝑥𝑦 ∈ L𝜙 ([𝑎, 𝑏] × [𝑐, 𝑑]). Moreover, in this case, the equality Var𝑅𝜙 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) 𝑏

𝑑

𝑏 𝑑

= ∫ 𝜙(|𝑓𝑥 (𝑥, 𝑐)|) 𝑑𝑥 + ∫ 𝜙(|𝑓𝑦 (𝑎, 𝑦)|) 𝑑𝑦 + ∫ ∫ 𝜙(|𝑓𝑥𝑦 (𝑥, 𝑦)|) 𝑑𝑦 𝑑𝑥 𝑎

𝑐

𝑎 𝑐

holds, where Var𝑅𝜙 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) denotes the 𝜙-variation (3.118) of 𝑓 in the sense of Riesz and Medvedev. Absolutely continuous functions not only play a prominent role in the theory of the Lebesgue integral, but are also important in integration techniques like the substi­

25 Here, 𝑓𝑥𝑦 ∈ L𝜙 ([𝑎, 𝑏] × [𝑐, 𝑑]) means of course that (𝜙 ∘ 𝑓𝑥𝑦 ) ∈ 𝐿 1 ([𝑎, 𝑏] × [𝑐, 𝑑]).

3.7 Comments on Chapter 3 | 259

tution formula²⁶ or the integration by parts formula. The latter is usually taught in courses on real functions in the following form: Theorem 3.52. For 𝑓, 𝑔 ∈ 𝐴𝐶([𝑎, 𝑏]), the formula 𝑏

𝑏

∫ 𝑓󸀠 (𝑥)𝑔(𝑥) 𝑑𝑥 = 𝑓(𝑏)𝑔(𝑏) − 𝑓(𝑎)𝑔(𝑎) − ∫ 𝑓(𝑥)𝑔󸀠 (𝑥) 𝑑𝑥 holds.

𝑎

(3.122)

𝑎

Note that both integrals in (3.122) make sense since the product of an 𝐿 1 -function and an 𝐿 ∞ -function is integrable. However, one can weaken this result in several aspects. For example, in [74, 75], the following result is given: Theorem 3.53. Suppose that two functions 𝑓, 𝑔 : [𝑎, 𝑏] → ℝ have the property that their product 𝑓𝑔 is absolutely continuous, and the derivatives 𝑓󸀠 and 𝑔󸀠 exist a.e. on [𝑎, 𝑏]. Moreover, assume that at least one of the functions 𝑓󸀠 𝑔 or 𝑓𝑔󸀠 is integrable. Then the formula (3.122) holds. In fact, under the hypotheses of Theorem 3.53, we know that the derivative (𝑓𝑔)󸀠 = 𝑓󸀠 𝑔 + 𝑓𝑔󸀠 exists a.e. on [𝑎, 𝑏] and is integrable. However, then both functions 𝑓󸀠 𝑔 and 𝑓𝑔󸀠 are integrable, and the assertion follows from the fundamental equality (3.31) for the Lebesgue integral. To see that Theorem 3.53 is not only formally more general than Theorem 3.52, we consider the following Example 3.54. Define 𝑓, 𝑔 : [0, 1] → ℝ by {sin 𝑓(𝑥) := { 0 {

1 𝑥

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 ,

{𝑥2 sin 𝑥1 𝑔(𝑥) := { 0 {

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 .

In other words, in the notation of (0.86), we have 𝑓 = 𝑓0,−1 and 𝑔 = 𝑓2,−1 . In Example 1.16, we have seen that the product function 𝑓𝑔 is Lipschitz continuous, and hence absolutely continuous. Moreover, the function {2𝑥 sin2 𝑥1 − 12 sin 𝑥2 for 0 < 𝑥 ≤ 1 , 𝑓(𝑥)𝑔󸀠 (𝑥) := { 0 for 𝑥 = 0 { is certainly integrable, being the sum of a continuous and a bounded function. Thus, all hypotheses of Theorem 3.53 are met, and formula (3.109) holds for 𝑓 and 𝑔. On the other hand, Theorem 3.52 does not apply since the function 𝑓 is not continuous, let alone absolutely continuous on [0, 1]. ♥ The phenomenon described in Example 3.54 is put into a more general framework in Exercise 3.64.

26 We will apply such formulas for change of variables in Section 5.3.

260 | 3 Absolutely continuous functions

3.8 Exercises to Chapter 3 We state some exercises on the topics covered in this chapter; exercises marked with an asterisk * are more difficult. Exercise 3.1. Show that 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) implies |𝑓| ∈ 𝐴𝐶([𝑎, 𝑏]). Exercise 3.2. Find a function 𝑓 ∈ ̸ 𝐴𝐶([0, 1]) such that |𝑓| ∈ 𝐴𝐶([0, 1]). Compare this with Exercise 1.4. Exercise 3.3. Suppose that 𝑓 ∈ 𝐶([𝑎, 𝑏]) and |𝑓| ∈ 𝐴𝐶([𝑎, 𝑏]). Show that 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]). Compare this with Exercise 1.5 and your example in Exercise 3.2. Exercise 3.4. In the notation of Exercise 0.70, does 𝑓, 𝑔 ∈ 𝐴𝐶([𝑎, 𝑏]) imply that 𝑓 ∨ 𝑔, 𝑓 ∧ 𝑔 ∈ 𝐴𝐶([𝑎, 𝑏])? Exercise 3.5. If 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]), show that |𝑓|𝑝 ∈ 𝐴𝐶([𝑎, 𝑏]) for 𝑝 ≥ 1. Is this also true for 0 < 𝑝 < 1? Exercise 3.6. For which sequences 𝐷 = (𝑑𝑛 )𝑛 is the zigzag function 𝑍𝐷 constructed in Definition 0.49 absolutely continuous on [0, 1]? Exercise 3.7. Let 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]). Prove that for every 𝜀 > 0, there exists a 𝛿 > 0 such that for all infinite collections {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]), the condition ∞

∑ |𝑏𝑘 − 𝑎𝑘 | ≤ 𝛿 𝑘=1

implies the condition

∞

∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| ≤ 𝜀 . 𝑘=1

Compare with Definition 3.1. Exercise 3.8. Given 𝑓 : [𝑎, 𝑏] → ℝ, suppose that for every 𝜀 > 0, there exists a 𝛿 > 0 with the following property: for each collection {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} of (not necessarily nonoverlapping!) subintervals of [𝑎, 𝑏], from 𝑛

∑ |𝑏𝑘 − 𝑎𝑘 | ≤ 𝛿, 𝑘=1

it follows that

𝑛

∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| ≤ 𝜀 . 𝑘=1

Prove that²⁷ 𝑓 ∈ 𝐿𝑖𝑝([𝑎, 𝑏]). Compare with Definition 3.1

27 Note that the converse is trivially true.

3.8 Exercises to Chapter 3

| 261

Exercise 3.9. Let 𝐶 be the Cantor set (3.3), and let ∞

[0, 1] \ 𝐶 = ⋃(𝑎𝑛 , 𝑏𝑛 ) 𝑛=1

be a representation of the complement of 𝐶 as a countable union of open intervals (which is possible since [0, 1] \ 𝐶 is open). Denote by 𝑐𝑛 := 12 (𝑎𝑛 + 𝑏𝑛) the corresponding midpoints. Given a positive sequence (𝛿𝑛 )𝑛 converging to zero, we define a function 𝑓 : [0, 1] → ℝ by for 𝑥 ∈ 𝐶 ,

0 { { { { { {𝛿𝑛 𝑓(𝑥) := { 𝑥−𝑎𝑛 { 𝛿𝑛 𝑐 −𝑎 { { { { 𝑏𝑛𝑛 −𝑥𝑛 {𝛿𝑛 𝑏𝑛 −𝑐𝑛

for 𝑥 = 𝑐𝑛 for 𝑎𝑛 < 𝑥 < 𝑐𝑛 , for 𝑐𝑛 < 𝑥 < 𝑏𝑛 .

Prove that 𝑓 ∈ 𝐶([0, 1]) ∩ 𝐿𝑢([0, 1]), and that the total variation of 𝑓 on [0, 1] is given by ∞

Var(𝑓; [0, 1]) = 2 ∑ 𝛿𝑛 . 𝑛=1

Use this to construct, by a suitable choice of 𝛿𝑛 , examples of functions 𝑓 ∈ 𝐴𝐶([0, 1]) and 𝑓 ∈ [𝐶([0, 1]) ∩ 𝐿𝑢([0, 1])] \ 𝐴𝐶([0, 1]). Exercise 3.10. Let 𝑓 : [𝑎, 𝑏] → ℝ be strictly increasing, and let 𝐷𝑒𝑟∞ (𝑓) := {𝑥 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑓󸀠 (𝑥) = ∞} . Prove that 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) if and only if 𝑓(𝐷𝑒𝑟∞ (𝑓)) is a nullset. Illustrate this result by means of the strict Cantor function (3.7). Exercise 3.11. Let 𝑓 : [𝑎, 𝑏] → [𝑐, 𝑑] be a homeomorphism, and let 𝐷𝑒𝑟0 (𝑓) be the set of critical points of 𝑓 defined in (3.102). Using Exercise 3.10, prove that 𝑓−1 ∈ 𝐴𝐶([𝑐, 𝑑]) if and only if 𝑓−1 (𝐷𝑒𝑟0 (𝑓)) is a nullset. Illustrate this result by means of the strict Cantor function (3.7). Exercise 3.12. Prove that the Cantor function (3.6) has the integral 1

∫ 𝜑(𝑥) 𝑑𝑥 = 0

1 . 2

What is the integral of the strict Cantor function (3.7)? Exercise 3.13. Let 𝐶 be the Cantor set (3.3), and let 𝑓 : 𝐶 → ℝ be defined as follows. Given 𝑥 ∈ 𝐶 with ternary representation (3.5), we put ∞

𝑓(𝑥) := ∑ 𝑥𝑘 3−𝑘! . 𝑘=1

Prove the following statements.

262 | 3 Absolutely continuous functions (a) From 𝑥, 𝑦 ∈ 𝐶 and 𝑥 − 𝑦 > 3−𝑛 , it follows that 𝑓(𝑥) − 𝑓(𝑦) ≥ 3−𝑛! . (b) The map 𝑓 : 𝐶 → [0, 1] is injective, but not surjective. (c) The map 𝑓 : 𝐶 → [0, 1] satisfies a Hölder condition |𝑓(𝑥) − 𝑓(𝑦)| ≤ 𝐿|𝑥 − 𝑦|𝛼

(𝑥, 𝑦 ∈ 𝐶)

for any 𝛼 ∈ (0, 1). Is 𝑓 absolutely continuous? Exercise 3.14. Prove that the Cantor function (3.6) belongs to 𝐿𝑖𝑝𝛼 ([0, 1]) for 𝛼 = log 2/ log 3, but does not belong to 𝐿𝑖𝑝𝛽 ([0, 1]) for any 𝛽 > log 2/ log 3. Exercise 3.15. Construct two functions 𝑔 ∈ 𝐶1 ([−1, 1]) and 𝑓 ∈ 𝐴𝐶([−1, 1])∩𝐶1 ([−1, 1]\ {0}) such that 𝑔 ∘ 𝑓 ∈ ̸ 𝐴𝐶([−1, 1]). Exercise 3.16. Suppose that 𝑓 : [𝑎, 𝑏] → ℝ satisfies a Hölder condition |𝑓(𝑥) − 𝑓(𝑦)| ≤ 𝐿|𝑥 − 𝑦|𝛼

(𝑎 ≤ 𝑥, 𝑦 ≤ 𝑏)

with some constant 𝐿 > 0. Which condition on 𝛼 then guarantees that 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏])? Exercise 3.17*. Let 𝑓 : [𝑎, 𝑏] → ℝ be absolutely continuous, and let 𝑔 : [𝑎, 𝑏] → ℝ be increasing. Suppose that 𝜆(𝑀𝑐 (𝑓)) = 𝜆(𝑀𝑐 (𝑔)), where 𝑀𝑐 is defined as in Exercise 0.1. Prove that 𝑔 is then also absolutely continuous. Exercise 3.18. Prove the inclusion 𝐿𝑖𝑝([𝑎, 𝑏]) ⊆ 𝐿𝑢([𝑎, 𝑏]), and give an example of a function 𝑓 ∈ 𝐿𝑢([0, 1]) \ 𝐿𝑖𝑝([0, 1]). Exercise 3.19. Is the inclusion 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) ⊆ 𝐿𝑢([𝑎, 𝑏]) also true for 𝛼 < 1? Compare with Exercise 3.8. Exercise 3.20. Suppose that 𝑓 : [𝑎, 𝑏] → ℝ is everywhere differentiable on [𝑎, 𝑏]. Prove that 𝑓 ∈ 𝐿𝑢([𝑎, 𝑏]). Compare with Exercise 3.18. Exercise 3.21. Given 𝑓, 𝑔 ∈ 𝐿𝑢([𝑎, 𝑏]), show that 𝑓 ⋅ 𝑔 ∈ 𝐿𝑢([𝑎, 𝑏]). Is it also true that 𝑓 + 𝑔 ∈ 𝐿𝑢([𝑎, 𝑏])? Exercise 3.22. Is the equality ⋃ 𝐴𝐶𝑝 ([𝑎, 𝑏]) = 𝐶([𝑎, 𝑏])

𝑝≥1

true? If not, construct a continuous function which does not belong to 𝐴𝐶𝑝 ([𝑎, 𝑏]) for any 𝑝 ≥ 1. Exercise 3.23. Does the Cantor function (3.6) belong to any of the spaces 𝐴𝐶𝑝 ([0, 1])? Exercise 3.24. Given 𝑝 ≥ 1, determine the precise values of 𝜃 for which the special zigzag function 𝑍𝜃 defined in (0.93) belongs to 𝐴𝐶𝑝 ([0, 1]).

3.8 Exercises to Chapter 3 |

263

Exercise 3.25. Define 𝑓 : [0, 1] → ℝ by 1/𝑞

{ 𝑥 sin2 𝑓(𝑥) := { log 𝑥 0 {

1 𝑥

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 .

Prove that 𝑓 ∈ 𝐴𝐶𝑞 ([0, 1]), but 𝑓 ∈ ̸ 𝑊𝐵𝑉𝑝 ([0, 1]) for 𝑝 < 𝑞. Exercise 3.26. Define 𝑓 : [0, 1] → ℝ by ∞

sin 𝑁𝑘 𝑥 , 𝑘/𝑞 𝑘=1 𝑁

𝑓(𝑥) := ∑

where 𝑁 ∈ ℕ. Prove that for 𝑁 sufficiently large, 𝑓 ∈ 𝐶([0, 1]) ∩ 𝑊𝐵𝑉𝑞 ([0, 1]), but 𝑓 ∈ ̸ 𝐴𝐶𝑞 ([0, 1]). Exercise 3.27. Given a bounded function 𝑓 : [𝑎, 𝑏] → ℝ, suppose that for every 𝜀 > 0, there exists a function 𝑔 ∈ 𝐴𝐶𝑝 ([𝑎, 𝑏]) such that } {𝑚 inf { ∑ Var𝑊 𝑝 (𝑓 − 𝑔; [𝑡𝑗−1 , 𝑡𝑗 ]) : {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏])} ≤ 𝜀 , } {𝑗=1 see (3.16). Prove that then also 𝑓 ∈ 𝐴𝐶𝑝 ([𝑎, 𝑏]). Exercise 3.28. Are the two norms (3.42) and (3.43) equivalent on 𝐴𝐶([𝑎, 𝑏])? Are they equivalent to the norm ‖𝑓‖𝐴𝐶 := |𝑓(𝑎)| + ‖𝑓󸀠 ‖𝐿 1 ? Exercise 3.29. Show that the space 𝐴𝐶([𝑎, 𝑏]) with any of the norms (3.42) or (3.43) is separable. Compare this with Exercise 1.49. Exercise 3.30. Use Theorem 3.20 to show that 𝐿𝑖𝑝([𝑎, 𝑏]) is dense in 𝐴𝐶([𝑎, 𝑏]) with respect to the norm (3.42). Exercise 3.31. Suppose that 𝑓 ∈ 𝐶([𝑎, 𝑏]) has the property that the set 𝑆 := {𝑥 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑓󸀠 (𝑥) does not exist} is countable, and 𝑓󸀠 (𝑥) ≡ 0 on [𝑎, 𝑏] \ 𝑆. Prove that 𝑓 is constant. Is the same true if 𝑆 is an uncountable nullset? Exercise 3.32. Let 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]), and let 𝑁 ⊂ [𝑎, 𝑏] be a nullset such that 𝑓󸀠 exists on [𝑎, 𝑏] \ 𝑁. Suppose that the restriction 𝑓󸀠 |[𝑎,𝑏]\𝑁 of 𝑓󸀠 to [𝑎, 𝑏] \ 𝑁 satisfies a Lipschitz condition with Lipschitz constant 𝐿. Prove that 𝑓󸀠 then exists on the whole interval [𝑎, 𝑏] (as a one-sided derivative at the boundary points 𝑎 and 𝑏) and satisfies a Lipschitz condition with Lipschitz constant 𝐿. Exercise 3.33. Is the inequality (3.32) in Theorem 3.19 always strict for 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏])\ 𝐴𝐶([𝑎, 𝑏])? Exercise 3.34. Illustrate Theorem 3.25 by means of the Cantor function 𝜑 ∈ 𝐵𝑉([0, 1])\ 𝐴𝐶([0, 1]) defined in (3.6).

264 | 3 Absolutely continuous functions Exercise 3.35. Carry out the details in the last part of Example 3.15. Exercise 3.36. Carry out the details in the last part of Example 3.23. Exercise 3.37. Illustrate Theorem 3.25 by means of the function 𝑔 = 𝜑󸀠 from Exam­ ple 3.16. Exercise 3.38*. Construct a function 𝑔 ∈ 𝐿 1 ([−1, 1]) such that 𝑔(0) = 0, ℎ

1 ∫ |𝑔(𝑡)| 𝑑𝑡 → ∞ (ℎ → 0) , ℎ 0

but the function 𝑓 : [−1, 1] → ℝ defined by 𝑥

𝑓(𝑥) := ∫ 𝑔(𝑡) 𝑑𝑡 −1 󸀠

is differentiable at 0 with 𝑓 (0) = 0. Exercise 3.39. Let 𝑓 : [𝑎, 𝑏] → ℝ be increasing, and let (3.38) be the decomposition of 𝑓 as a sum of an absolutely continuous function 𝑓ac and a singular function 𝑓sg . Show that both 𝑓ac and 𝑓sg are then also increasing. Exercise 3.40. We define a sequence of “sawtooth functions” as follows. For 0 ≤ 𝑥 ≤ 1, we first put 𝑓1 (𝑥) := 12 − |𝑥 − 12 |. For 𝑛 = 2, 3, 4, . . ., we then define 𝑓𝑛 on [0, 2−𝑛+1 ] by 𝑓𝑛 (𝑥) := 2−𝑛 − |𝑥 − 2−𝑛| and extend 𝑓𝑛 periodically to the whole interval [0, 1]. Prove the following properties of the sequence (𝑓𝑛 )𝑛 . (a) The sequence (𝑓𝑛 )𝑛 converges uniformly on [0, 1] to 𝑓(𝑥) ≡ 0. (b) All functions 𝑓𝑛 have the same graph length 𝐿(𝛤(𝑓𝑛 ); [0, 1]) = √2. (c) The convergence 𝐿(𝛤(𝑓𝑛 )) → 𝐿(𝛤(𝑓)) does not hold as 𝑛 → ∞. Also, calculate the total variations Var(𝑓𝑛 ; [0, 1]) and Var(𝑓; [0, 1]) and comment on the result. Exercise 3.41. Let 𝑓 : [0, 1] → [0, 1] be monotonically increasing with 𝑓(0) = 0 and 𝑓(1) = 1. Show that the graph length of 𝑓 satisfies √2 ≤ 𝐿(𝛤(𝑓)) ≤ 2. Moreover, prove that 𝐿(𝛤(𝑓)) = √2 if and only if 𝑓(𝑥) = 𝑥, and 𝐿(𝛤(𝑓)) = 2 if and only if 𝑓󸀠 (𝑥) ≡ 0 a.e. on [0, 1]. Exercise 3.42. Calculate 𝐿(𝛤(𝑓)) for 𝑓(𝑥) = 𝑥2 on [0, 1]. Exercise 3.43. Let 𝑓 : [𝑎, 𝑏] → ℝ be differentiable with (Riemann) integrable deriva­ tive 𝑓󸀠 on [𝑎, 𝑏]. Prove that then (3.60) holds, and compare with Example 3.15. Exercise 3.44. For 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]), let 𝑉𝑓 be the variation function (1.13) of 𝑓 and 𝐿 𝑓 be the length function (3.57) of 𝑓. Prove that 𝐿 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) and Var(𝐿 𝑓 ; [𝑎, 𝑏]) = 𝐿(𝛤(𝑉𝑓 ); [𝑎, 𝑏]) ,

3.8 Exercises to Chapter 3 |

265

i.e. the total variation of the length function is the length of the graph of the variation function. Exercise 3.45. Is the inequality (3.64) in Theorem 3.31 always strict for 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏])\ 𝐴𝐶([𝑎, 𝑏])? Exercise 3.46. Define 𝑔 : [0, 1] → ℝ by 1 {ent ( ent(1/𝑥) ) 𝑔(𝑥) := { 0 {

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 ,

where ent(𝜉) denotes the integer part of 𝜉. Show that 𝑔 ∈ 𝐵𝑉([0, 1]) \ 𝐴𝐶([0, 1]) and calculate Var(𝑔; [0, 1]) and 𝐿(𝛤(𝑔); [0, 1]). Exercise 3.47. Determine all 𝑝 ∈ [1, ∞) for which the function from Definition 0.47 belongs to 𝑅𝐵𝑉𝑝 ([0, 1]). Exercise 3.48*. Let 1 < 𝑝 < ∞, 𝑘 ≥ 2, and 𝑓 ∈ 𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]), see Definition 3.38. Prove that the derivative 𝑓(𝑘−2) belongs then to 𝑅𝐵𝑉2,1 ([𝑎, 𝑏]). Conclude that the right and left derivative 𝑓+(𝑘−1) and 𝑓−(𝑘−1) exist on [𝑎, 𝑏] and are, respectively, right and left continuous. Exercise 3.49. Prove the (strict) inclusions 𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]) ⊃ 𝑅𝐵𝑉𝑘+1,𝑝 ([𝑎, 𝑏])

(𝑘 = 1, 2, 3, . . . ; 1 ≤ 𝑝 < ∞)

and 𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]) ⊂ 𝑅𝐵𝑉𝑘,𝑞 ([𝑎, 𝑏]) (𝑘 = 1, 2, 3, . . . ; 1 ≤ 𝑝 < 𝑞 < ∞) and compare with Proposition 2.73. Exercise 3.50. Prove the continuous imbedding 𝑅𝐵𝑉𝑘+1,𝑝 ([𝑎, 𝑏]) 󳨅→ 𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]) (𝑘 ∈ ℕ, 1 ≤ 𝑝 < ∞) , where 𝑅𝐵𝑉𝑘,𝑝 ([𝑎, 𝑏]) is the space defined by the norm (3.77), and calculate the sharp imbedding constant 𝑐(𝑅𝐵𝑉𝑘+1 , 𝑅𝐵𝑉𝑘 ). Exercise 3.51. Let 𝜙 and 𝜓 be two Young functions which satisfy condition ∞1 and 𝜓(𝑡) ≤ 𝜙(𝑐𝑡)

(𝑡 ≥ 𝑇)

for some constants 𝑐 > 0 and 𝑇 > 0. Prove the continuous imbedding 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) 󳨅→ 𝑅𝐵𝑉𝜓 ([𝑎, 𝑏]) , where 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) is the space defined by the norm (2.99),

266 | 3 Absolutely continuous functions Exercise 3.52. Prove the continuous imbedding 𝑅𝐵𝑉𝑘,𝜙 ([𝑎, 𝑏]) 󳨅→ 𝑅𝐵𝑉𝑘,1 ([𝑎, 𝑏]) (𝑘 ∈ ℕ) , where 𝑅𝐵𝑉𝑘,𝜙 ([𝑎, 𝑏]) is the space defined by the norm (3.86), and 𝜙 is a Young function satisfying condition ∞1 . Exercise 3.53. The set 𝐴𝐶1 ([𝑎, 𝑏]) consists of all differentiable functions with an ab­ solutely continuous derivative on [𝑎, 𝑏]. Show that 𝐴𝐶1 ([𝑎, 𝑏]), equipped with the norm (0.44), is a Banach space. Also, show that 𝐴𝐶1 󳨅→ 𝐿𝑖𝑝 and calculate the sharp imbed­ ding constant 𝑐(𝐴𝐶1 , 𝐿𝑖𝑝), see (0.36). Exercise 3.54. The space 𝐴𝐶([𝑎, 𝑏]) can be imbedded in 𝐴𝐶(ℝ) by extending each 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) to be constant outside [𝑎, 𝑏]. Prove that a closed bounded set 𝑀 ⊂ 𝐴𝐶([𝑎, 𝑏]) is compact in (𝐴𝐶([𝑎, 𝑏]), ‖ ⋅ ‖𝐴𝐶 ) if and only if both lim sup Var(𝑓; ℝ \ (−𝑛, 𝑛)) = 0

𝑛→∞

𝑓∈𝑀

and lim sup ‖𝑓 − 𝑓𝜏 ‖𝐴𝐶 = 0 ,

𝜏→0 𝑓∈𝑀

where 𝑓𝜏 denotes the shift 𝑓𝜏 (𝑡) := 𝑓(𝑡 + 𝜏) of 𝑓. Exercise 3.55. Using Exercise 3.33 (but not general facts from functional analysis), show that the closed unit ball in (𝐴𝐶([𝑎, 𝑏]), ‖ ⋅ ‖𝐴𝐶 ) is not compact. Exercise 3.56. Find examples of functions 𝑓, 𝑔 ∈ 𝐵𝑉([0, 1]) for which the first and last inequalities in (3.99) are strict. Exercise 3.57. Calculate 𝑑𝑘 (𝜑, 𝜓) (𝑘 = 1, 2, 3, 4) for the four metrics (3.95)–(3.98), where 𝜑 is the Cantor function (3.6) and 𝜓 is the strict Cantor function (3.7), and check the estimates (3.99). Exercise 3.58. Given a countable set 𝑀 = {𝑦1 , 𝑦2 , 𝑦3 , . . .}, define 𝑓 : [0, 1] → ℝ by {𝑦𝑘 𝑓(𝑥) := { linear {

for

1 𝑘

−

1 2𝑘(𝑘+1)

≤𝑥≤

1 𝑘

+

1 2𝑘(𝑘+1)

,

otherwise .

Prove that 𝑓 ∈ 𝐶([0, 1]) and 𝑓(𝐸𝑥𝑡(𝑓)) = 𝑀, see (3.105). Exercise 3.59. Suppose that 𝑓 : [𝑎, 𝑏] → ℝ satisfies 𝐸𝑥𝑡(𝑓) = [𝑎, 𝑏] and 𝑓(𝐸𝑥𝑡(𝑓)) = {𝑐, 𝑑} for 𝑐 ≠ 𝑑, see (3.104) and (3.105). Prove that 𝑓 is discontinuous everywhere on [𝑎, 𝑏]. Exercise 3.60. Suppose that 𝑓 ∈ 𝐶([𝑎, 𝑏]) satisfies 𝐸𝑥𝑡(𝑓) = [𝑎, 𝑏], see (3.104). Does it follow that 𝑓 is constant?

3.8 Exercises to Chapter 3

| 267

Exercise 3.61. For 0 < 𝛼 < 1, let 𝑓𝛼 : [0, 1] × [0, 1] → ℝ be defined by 𝑓𝛼 (𝑥, 𝑦) := (𝑥 − 𝑦)𝛼 . Use Theorem 3.51 to determine all 𝑝 ≥ 1 such that 𝑓𝛼 ∈ 𝑅𝐵𝑉𝑝 ([0, 1] × [0, 1]). Also, calculate ‖𝑓𝛼 ‖𝑅𝐵𝑉𝑝 in this case, and compare with Example 3.35. Exercise 3.62. Prove that 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏] × [𝑐, 𝑑]), see Definition 3.50, if and only if 𝑓(𝑎, ⋅) ∈ 𝐴𝐶([𝑐, 𝑑]), 𝑓(⋅, 𝑦) ∈ 𝐴𝐶([𝑎, 𝑏]) for every 𝑦 ∈ [𝑐, 𝑑], 𝑓𝑥 (𝑥, ⋅) ∈ 𝐴𝐶([𝑐, 𝑑]) for almost every 𝑥 ∈ [𝑎, 𝑏], and 𝑓𝑥𝑦 ∈ 𝐿 1 ([𝑎, 𝑏] × [𝑐, 𝑑]). Exercise 3.63*. Given 𝑔 ∈ 𝐿 1 ([𝑎, 𝑏]), define 𝑓 as in (3.20) and put 󵄨󵄨 𝑔(𝑥 + ℎ) − 𝑔(𝑥) 󵄨󵄨 󵄨 󵄨󵄨 𝐸𝑐 (𝑔) := {𝑥 : 𝑎 < 𝑥 < 𝑏, lim sup 󵄨󵄨󵄨 󵄨󵄨 > 𝑐} 󵄨 󵄨󵄨 ℎ 󵄨 ℎ→0 Prove that 𝜆(𝐸𝑐 (𝑔)) ≤

(𝑐 > 0) .

4 ‖𝑔‖𝐿 1 𝑐

and compare with Exercise 3.10. Exercise 3.64. For 𝛼, 𝛽, 𝛾, 𝛿 ∈ ℝ, define 𝑓, 𝑔 : [0, 1] → ℝ by {𝑥𝛼 sin 𝑥𝛽 𝑓(𝑥) := { 0 {

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 ,

{𝑥𝛾 sin 𝑥𝛿 𝑔(𝑥) := { 0 {

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0 .

Using Table 2.4 in Chapter 2 to determine all values of 𝛼, 𝛽, 𝛾, and 𝛿 for which Theorem 3.53 applies to 𝑓 and 𝑔, but Theorem 3.52 does not. Exercise 3.65. Formulate and prove an analogue to Theorem 3.46 for 𝑓 ∈ 𝐿𝑖𝑝([𝑎, 𝑏]), and illustrate this by means of the function 𝑔 from Example 3.45. Exercise 3.66. A continuous function 𝑓 : [𝑎, 𝑏] → ℝ is called piecewise linear if there exists a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) such that 𝑓 is affine on each interval [𝑡𝑗−1 , 𝑡𝑗 ]. Prove that every piecewise linear function is absolutely continuous and cal­ culate its norm (3.42). Exercise 3.67. Denote by 𝑃𝐿([𝑎, 𝑏]) the set of all piecewise linear functions on [𝑎, 𝑏], see Exercise 3.66. Show that 𝑃𝐿([𝑎, 𝑏]) = 𝐽(𝑆([𝑎, 𝑏])), where 𝑆([𝑎, 𝑏]) denotes the set of step functions introduced in Section 0.4, and 𝐽 denotes the linear integral operator (3.107). Exercise 3.68. Use the result of Exercise 3.67 and the fact that 𝑆([𝑎, 𝑏]) is dense in 𝐿 1 ([𝑎, 𝑏]) to prove that 𝑃𝐿([𝑎, 𝑏]) is dense in 𝐴𝐶([𝑎, 𝑏]) with respect to the norm (3.43).

4 Riemann–Stieltjes integrals By means of the classical Riemann–Stieltjes integral, it is possible to construct an isometry between the dual space of the Chebyshev space 𝐶([𝑎, 𝑏]) and a subspace of (suitably regularized) functions in 𝐵𝑉([𝑎, 𝑏]). Similarly, an analogous isometry makes it possible to identify the dual space of the Lebesgue space 𝐿 𝑝 ([𝑎, 𝑏]) (1 < 𝑝 < ∞) with the space 𝑅𝐵𝑉𝑝/(𝑝−1) ([𝑎, 𝑏]), where 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) denotes the space of functions of bounded 𝑝-variation in Riesz’s sense introduced in Chapter 2. Here, one does not need regularizations since all functions in 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) are continuous for 𝑝 > 1. One may consider the same isometry also for the case 𝑝 = 1 which leads to the space 𝐿𝑖𝑝([𝑎, 𝑏]) of Lipschitz continuous functions on [𝑎, 𝑏], but not for the case 𝑝 = ∞ which has to be treated differently. In the last section, we show how to extend the Riemann–Stieltjes integral to the case when the “integrator” does not necessarily belong to the classical space 𝐵𝑉([𝑎, 𝑏]), but to the larger space 𝛷𝐵𝑉([𝑎, 𝑏]) of functions of bounded Schramm variation introduced in Section 2.3.

4.1 Classical RS-integrals In this section, we start with the definition of the classical Riemann–Stieltjes integral and discuss some of its properties. To begin with, let 𝛼 : [𝑎, 𝑏] → ℝ be monotonically increasing, and let 𝑓 : [𝑎, 𝑏] → ℝ be bounded. Given a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), consider the numbers 𝑀𝑗 := sup {𝑓(𝑥) : 𝑡𝑗−1 ≤ 𝑥 ≤ 𝑡𝑗 },

𝑚𝑗 := inf {𝑓(𝑥) : 𝑡𝑗−1 ≤ 𝑥 ≤ 𝑡𝑗 }

(4.1)

for 𝑗 = 1, 2, . . . , 𝑚. As usual, we will call the real number 𝑚

𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) := ∑ 𝑚𝑗 (𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )),

(4.2)

𝑗=1

the lower Riemann–Stieltjes sum (or lower RS-sum) of 𝑓 with respect to 𝑃 and 𝛼, and the real number 𝑚

𝑈𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) := ∑ 𝑀𝑗 (𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )),

(4.3)

𝑗=1

the upper Riemann–Stieltjes sum (or upper RS-sum) of 𝑓 with respect to 𝑃 and 𝛼. A useful monotonicity property of these sums is given in the following Proposition 4.1. For 𝑃, 𝑄 ∈ P([𝑎, 𝑏]) with 𝑃 ⊆ 𝑄, the estimates 𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) ≤ 𝐿 𝛼 (𝑓, 𝑄; [𝑎, 𝑏]) ≤ 𝑈𝛼 (𝑓, 𝑄; [𝑎, 𝑏]) ≤ 𝑈𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) hold.

(4.4)

4.1 Classical RS-integrals

| 269

Proof. Let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), and let 𝑄 = {𝑡0 , . . . , 𝑡𝑖−1 , 𝑡∗ , 𝑡𝑖 , . . . , 𝑡𝑚 } be a partition consisting of 𝑃 and one additional point 𝑡∗ ∈ (𝑡𝑖−1 , 𝑡𝑖 ). Then we have 𝑖−1

𝐿 𝛼 (𝑓, 𝑄; [𝑎, 𝑏]) = ∑ 𝑚𝑗 (𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )) + 𝑚∗𝑖−1 (𝛼(𝑡∗ ) − 𝛼(𝑡𝑖−1 )) 𝑗=1

𝑚

+ 𝑚∗𝑖 (𝛼(𝑡𝑖 ) − 𝛼(𝑡∗ )) + ∑ 𝑚𝑗 (𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )) , 𝑗=𝑖+1

where 𝑚∗𝑖−1 := inf {𝑓(𝑥) : 𝑡𝑖−1 ≤ 𝑥 ≤ 𝑡∗ }

𝑚∗𝑖 := inf {𝑓(𝑥) : 𝑡∗ ≤ 𝑥 ≤ 𝑡𝑖 } .

Obviously, 𝑚∗𝑖−1 ≥ 𝑚𝑖 and 𝑚∗𝑖 ≥ 𝑚𝑖 , and hence 𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) ≤ 𝐿 𝛼 (𝑓, 𝑃 ∪ {𝑡∗ }; [𝑎, 𝑏]) . By applying induction on the number of added points, we derive the first estimate in (4.4). The third estimate is proved in the same way, while the second estimate is of course trivial. Definition 4.2. We say that a function 𝑓 ∈ 𝐵([𝑎, 𝑏]) is Riemann–Stieltjes integrable (or RS-integrable, for short) with respect to 𝛼 on [𝑎, 𝑏] if for each 𝜀 > 0, there exists a partition 𝑃 ∈ P([𝑎, 𝑏]) such that 𝑈𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) − 𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) ≤ 𝜀 . In this case, we write 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]).

(4.5) ◼

Thus, the condition 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) means that the equality inf {𝑈𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) : 𝑃 ∈ P([𝑎, 𝑏])} = sup {𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) : 𝑃 ∈ P([𝑎, 𝑏])}

(4.6)

holds. In this case, we denote the common value in (4.6) by 𝑏

𝑏

𝐼 = ∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = ∫ 𝑓 𝑑𝛼 𝑎

(4.7)

𝑎

and call it the Riemann–Stieltjes integral (or RS-integral, for short) of 𝑓 with respect to 𝛼 on [𝑎, 𝑏]. The function 𝑓 in (4.7) is often called the integrand and the function 𝛼 the integrator. Obviously, in case 𝛼(𝑥) = 𝑥, Definition 4.2 gives nothing other than Riemann integrability, and (4.7) is the classical Riemann integral of 𝑓. In this case, we simply write¹ 𝑓 ∈ 𝑅𝑆([𝑎, 𝑏]). We point out, however, that, in general, the function 𝛼 need not even be continuous. In the next proposition, we collect some useful properties of the RS-integral.

1 Of course, we had better write 𝑅([𝑎, 𝑏]) instead of 𝑅𝑆([𝑎, 𝑏]) for Riemann integrable functions; how­ ever, this could be confused with the space of regular functions which we introduced in Chapter 0.

270 | 4 Riemann–Stieltjes integrals Proposition 4.3. Let 𝛼, 𝛽 : [𝑎, 𝑏] → ℝ be monotonically increasing, 𝑓, 𝑔 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]), and 𝜇 ∈ ℝ. Then the following is true. (a) The integral (4.7) is additive with respect to the integrand, i.e.² 𝑏

𝑏

𝑏

∫(𝑓 + 𝑔) 𝑑𝛼 = ∫ 𝑓 𝑑𝛼 + ∫ 𝑔 𝑑𝛼 . 𝑎

𝑎

(4.8)

𝑎

(b) The integral (4.7) is additive with respect to the integrator, i.e.³ 𝑏

𝑏

𝑏

∫ 𝑓 𝑑(𝛼 + 𝛽) = ∫ 𝑓 𝑑𝛼 + ∫ 𝑓 𝑑𝛽 . 𝑎

𝑎

(4.9)

𝑎

(c) The integral (4.7) is homogeneous with respect to the integrand, i.e. 𝑏

𝑏

∫(𝜇𝑓) 𝑑𝛼 = 𝜇 ∫ 𝑓 𝑑𝛼 𝑎

(4.10)

𝑎

for 𝜇 ∈ ℝ. (d) The integral (4.7) is homogeneous with respect to the integrator, i.e. 𝑏

𝑏

∫ 𝑓 𝑑(𝜇𝛼) = 𝜇 ∫ 𝑓 𝑑𝛼(𝑥) 𝑎

(4.11)

𝑎

for 𝜇 > 0. (e) The integral (4.7) is monotone with respect to the integrand, which means that 𝑓(𝑥) ≤ 𝑔(𝑥) on [𝑎, 𝑏] implies 𝑏

𝑏

∫ 𝑓 𝑑𝛼 ≤ ∫ 𝑔 𝑑𝛼 . 𝑎

(4.12)

𝑎

In particular, 𝑏

𝑚(𝑓)[𝛼(𝑏) − 𝛼(𝑎)] ≤ ∫ 𝑓 𝑑𝛼 ≤ 𝑀(𝑓)[𝛼(𝑏) − 𝛼(𝑎)] ,

(4.13)

𝑎

where 𝑚(𝑓) is defined in (0.61) and 𝑀(𝑓) in (0.62).

2 The statement (a) is meant in the following sense: from 𝑓, 𝑔 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]), it follows that also 𝑓 + 𝑔 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]), and (4.8) holds. 3 The statement (b) is meant in the following sense: from 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) ∩ 𝑅𝑆𝛽 ([𝑎, 𝑏]), it follows that also 𝑓 ∈ 𝑅𝑆𝛼+𝛽 ([𝑎, 𝑏]), and (4.9) holds.

4.1 Classical RS-integrals

| 271

Proof. In the sequel, we drop the interval [𝑎, 𝑏] in (4.2) and (4.3) and the argument 𝑥 in (4.7) so as not to overburden the notation. (a) For every 𝑃 ∈ P([𝑎, 𝑏]), we have 𝐿 𝛼 (𝑓, 𝑃) + 𝐿 𝛼 (𝑔, 𝑃) ≤ 𝐿 𝛼 (𝑓 + 𝑔, 𝑃) ≤ 𝑈𝛼 (𝑓 + 𝑔, 𝑃) ≤ 𝑈𝛼 (𝑓, 𝑃) + 𝑈𝛼 (𝑔, 𝑃) .

(4.14)

From 𝑓, 𝑔 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]), it follows that for each 𝜀 > 0, we can find partitions 𝑃𝑓 , 𝑃𝑔 ∈ P([𝑎, 𝑏]) such that 𝑈𝛼 (𝑓, 𝑃𝑓 ) − 𝐿 𝛼 (𝑓, 𝑃𝑓 ) ≤ 𝜀,

𝑈𝛼 (𝑔, 𝑃𝑔 ) − 𝐿 𝛼 (𝑔, 𝑃𝑔 ) ≤ 𝜀 .

Adding side by side of the above inequalities, we get 𝑈𝛼 (𝑓, 𝑃𝑓 ) + 𝑈𝛼 (𝑔, 𝑃𝑔 ) − 𝐿 𝛼 (𝑓, 𝑃𝑓 ) − 𝐿 𝛼 (𝑔, 𝑃𝑔 ) ≤ 2𝜀 . Taking now into account (4.14) and Proposition 4.1, we obtain 𝐿 𝛼 (𝑓, 𝑃𝑓 ) + 𝐿 𝛼 (𝑔, 𝑃𝑔 ) ≤ 𝐿 𝛼 (𝑓 + 𝑔, 𝑃𝑓 ∪ 𝑃𝑔 ) ≤ 𝑈𝛼 (𝑓 + 𝑔, 𝑃𝑓 ∪ 𝑃𝑔 ) ≤ 𝑈𝛼 (𝑓, 𝑃𝑓 ) + 𝑈𝛼 (𝑔, 𝑃𝑔 ) . Consequently, if we denote by 𝑃 the common refinement 𝑃𝑓 ∪ 𝑃𝑔 of 𝑃𝑓 and 𝑃𝑔 , we conclude that 𝑈𝛼 (𝑓 + 𝑔, 𝑃) − 𝐿 𝛼 (𝑓 + 𝑔, 𝑃) ≤ 2𝜀 , which shows that 𝑓 + 𝑔 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]). For the same partition 𝑃, we further have 𝑏

𝑏

𝑈𝛼 (𝑓, 𝑃) ≤ ∫ 𝑓 𝑑𝛼 + 𝜀,

𝑈𝛼 (𝑔, 𝑃) ≤ ∫ 𝑔 𝑑𝛼 + 𝜀 .

𝑎

𝑎

Consequently, (4.14) implies that 𝑏

𝑏

𝑏

∫(𝑓 + 𝑔) 𝑑𝛼 ≤ 𝑈𝛼 (𝑓 + 𝑔, 𝑃) ≤ ∫ 𝑓 𝑑𝛼 + ∫ 𝑔 𝑑𝛼 + 2𝜀 𝑎

and so

𝑎

𝑏

𝑏

𝑎

𝑏

∫(𝑓 + 𝑔) 𝑑𝛼 ≤ ∫ 𝑓 𝑑𝛼 + ∫ 𝑔 𝑑𝛼 𝑎

𝑎

𝑎

since 𝜀 > 0 was arbitrary. Replacing 𝑓 by −𝑓 and 𝑔 by −𝑔, we obtain the reverse esti­ mate, and so we have proved (4.8). The assertion (b) is proved similarly, building on the inequalities 𝐿 𝛼 (𝑓, 𝑃) + 𝐿 𝛽 (𝑓, 𝑃) ≤ 𝐿 𝛼+𝛽 (𝑓, 𝑃) ≤ 𝑈𝛼+𝛽 (𝑓, 𝑃) ≤ 𝑈𝛼 (𝑓, 𝑃) + 𝑈𝛽 (𝑓, 𝑃) for every 𝑃 ∈ P([𝑎, 𝑏]).

272 | 4 Riemann–Stieltjes integrals The assertion (c) follows from the equalities 𝐿 𝛼 (𝜇𝑓, 𝑃) = 𝜇𝐿 𝛼 (𝑓, 𝑃),

𝑈𝛼 (𝜇𝑓, 𝑃) = 𝜇𝑈𝛼 (𝑓, 𝑃) ,

while the assertion (d) follows from the equalities 𝐿 𝜇𝛼 (𝑓, 𝑃) = 𝜇𝐿 𝛼 (𝑓, 𝑃),

𝑈𝜇𝛼 (𝑓, 𝑃) = 𝜇𝑈𝛼 (𝑓, 𝑃) .

To prove (e), it suffices to observe that 𝑓(𝑥) ≤ 𝑔(𝑥) on [𝑎, 𝑏] clearly implies both 𝐿 𝛼 (𝑓, 𝑃) ≤ 𝐿 𝛼 (𝑔, 𝑃) and 𝑈𝛼 (𝑓, 𝑃) ≤ 𝑈𝛼 (𝑔, 𝑃). The estimate (4.13) is a consequence of (4.12). Before discussing several classes of RS-integrable functions, we want to extend Defi­ nition 4.2 to functions 𝛼 of bounded variation. Proposition 4.3 (b) suggests how to do this: given 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), consider any decomposition 𝛼 = 𝛽 − 𝛾 of 𝛼 as the differ­ ence of two monotonically increasing functions 𝛽, 𝛾 : [𝑎, 𝑏] → ℝ (for example, the Jordan decomposition described in Theorem 1.5). Then we define the RS-integral of 𝑓 ∈ 𝑅𝑆𝛽 ([𝑎, 𝑏]) ∩ 𝑅𝑆𝛾 ([𝑎, 𝑏]) in the natural way by 𝑏

𝑏

𝑏

∫ 𝑓 𝑑𝛼 := ∫ 𝑓 𝑑𝛽 − ∫ 𝑓 𝑑𝛾 . 𝑎

𝑎

(4.15)

𝑎

Proposition 4.3 (b) guarantees that this definition does not depend on the choice of 𝛽 and 𝛾. The Riemann integral may be introduced not only by means of upper and lower sums, but also with so-called Riemann sums in which the numbers 𝑀𝑗 and 𝑚𝑗 in (4.1) are replaced by the value of 𝑓 at an arbitrary intermediate points. The same is possible for the Riemann–Stieltjes integral. Definition 4.4. Let 𝛼 : [𝑎, 𝑏] → ℝ be monotonically increasing, and let 𝑓 : [𝑎, 𝑏] → ℝ be bounded. Given a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), choose a set 𝛱 = {𝜏1 , 𝜏2 , . . . , 𝜏𝑚 } of points satisfying 𝑎 = 𝑡0 ≤ 𝜏1 ≤ 𝑡1 ≤ . . . ≤ 𝑡𝑚−1 ≤ 𝜏𝑚 ≤ 𝑡𝑚 = 𝑏 .

(4.16)

Then we will call the sum 𝑚

𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) := ∑ 𝑓(𝜏𝑗 )(𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 ))

(4.17)

𝑗=1

the Riemann–Stieltjes sum (or 𝑅𝑆-sum, for short) in what follows.

◼

By definition, the notation lim 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) =: 𝐴 ,

𝜇(𝑃)→0

(4.18)

4.1 Classical RS-integrals

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273

where 𝜇(𝑃) denotes the mesh size (1.2) of 𝑃, then has the following meaning: for any 𝜀 > 0, there exists a 𝛿 > 0 such that |𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) − 𝐴| ≤ 𝜀

(4.19)

for any partition 𝑃 satisfying 𝜇(𝑃) ≤ 𝛿 and any set 𝛱 of intermediate points⁴ satisfying (4.16). Proposition 4.5. If the limit (4.18) exists, then 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) and 𝑏

lim 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) = ∫ 𝑓(𝑥) 𝑑𝛼(𝑥) ,

𝜇(𝑃)→0

(4.20)

𝑎

i.e. the RS-integral of 𝑓 with respect to 𝛼 coincides with this limit. Proof. Denote the limit as in (4.18) by 𝐴, and let 𝜀 > 0. By definition, we find a 𝛿 > 0 such that 𝐴 − 𝜀 ≤ 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) ≤ 𝐴 + 𝜀 (4.21) for any partition 𝑃 satisfying 𝜇(𝑃) ≤ 𝛿. If we let the intermediate points 𝜏𝑗 vary over the intervals [𝑡𝑗−1 , 𝑡𝑗 ] (𝑗 = 1, 2, . . . , 𝑚) and take the corresponding infimum and supremum over 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]), from (4.21), we get 𝐴 − 𝜀 ≤ 𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) ≤ 𝑈𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) ≤ 𝐴 + 𝜀 . By Definition 4.2, this means nothing more than 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) and 𝑏

∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = 𝐴 𝑎

since the real number 𝐴 in (4.18) is uniquely determined. We observe that the converse implication of the statement of Proposition 4.5 is also true: for every 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]), the limit (4.18) exists and coincides with the RS-integral of 𝑓 with respect to 𝛼. Indeed, this is a simple consequence of the inequalities 𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) ≤ 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) ≤ 𝑈𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) which holds for increasing 𝛼 and all sets 𝛱 = {𝜏1 , 𝜏2 , . . . , 𝜏𝑚 } satisfying (4.16). Now, we state some useful criteria for the existence of an RS-integral. The follow­ ing criterion is a consequence of Proposition 4.5.

4 Observe that 𝜇(𝑃) → 0 implies 𝜇(𝛱) → 0, and so the relation (4.18) is actually independent of 𝛱. For this reason, we will sometimes write 𝑆𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) instead of 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) whenever we consider the limit 𝜇(𝑃) → 0 in the sequel.

274 | 4 Riemann–Stieltjes integrals Corollary 4.6. Let 𝛼 : [𝑎, 𝑏] → ℝ be monotonically increasing. Then 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) if and only if for each 𝜀 > 0, there exists 𝛿 > 0 such that for arbitrary partitions 𝑃1 , 𝑃2 ∈ P([𝑎, 𝑏]) with 𝜇(𝑃𝑖 ) ≤ 𝛿, we have |𝑆𝛼 (𝑓, 𝑃1 , 𝛱1 ; [𝑎, 𝑏]) − 𝑆𝛼 (𝑓, 𝑃2 , 𝛱2 ; [𝑎, 𝑏])| ≤ 𝜀 ,

(4.22)

where 𝛱1 and 𝛱2 are arbitrary sets of intermediate points of 𝑃1 and 𝑃2 , respectively. Proof. The statement follows immediately from the Cauchy criterion for the existence of the limit (4.18). We may derive from Corollary 4.6 another useful property of the RS-integral which is often used in explicit calculations. Proposition 4.7. From 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]), it follows that 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑐]) and 𝑓 ∈ 𝑅𝑆𝛼 ([𝑐, 𝑏]) for each 𝑐 ∈ (𝑎, 𝑏); moreover, the equality 𝑏

𝑐

𝑏

∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = ∫ 𝑓(𝑥) 𝑑𝛼(𝑥) + ∫ 𝑓(𝑥) 𝑑𝛼(𝑥) 𝑎

𝑎

(4.23)

𝑐

holds in this case. Proof. Given 𝜀 > 0, by Corollary 4.6, we find 𝛿 > 0 such that (4.22) holds for arbitrary partitions 𝑃1 , 𝑃2 ∈ P([𝑎, 𝑏]) with 𝜇(𝑃𝑖 ) ≤ 𝛿 and arbitrary sets 𝛱1 and 𝛱2 of intermediate points of 𝑃1 and 𝑃2 , respectively.⁵ Fix 𝑃1󸀠 , 𝑃2󸀠 ∈ P([𝑎, 𝑐]) with 𝜇(𝑃1󸀠 ) ≤ 𝛿 and 𝜇(𝑃2󸀠 ) ≤ 𝛿, and corresponding sets 𝛱1󸀠 and 󸀠 𝛱2 of intermediate points. Moreover, choose any 𝑃󸀠󸀠 ∈ P([𝑐, 𝑏]) with 𝜇(𝑃󸀠󸀠 ) ≤ 𝛿 and a corresponding set 𝛱󸀠󸀠 of intermediate points. Then 𝑃𝑖 := 𝑃𝑖󸀠 ∪ 𝑃󸀠󸀠 ∈ P([𝑎, 𝑏]) satisfies 𝜇(𝑃𝑖 ) ≤ 𝛿 (𝑖 = 1, 2), and thus by applying the above condition, we get |𝑆𝛼 (𝑓, 𝑃1󸀠 , 𝛱1󸀠 ; [𝑎, 𝑐]) − 𝑆𝛼 (𝑓, 𝑃2󸀠 , 𝛱2󸀠 ; [𝑎, 𝑐])| = |𝑆𝛼 (𝑓, 𝑃1 , 𝛱1 ; [𝑎, 𝑏]) − 𝑆𝛼 (𝑓, 𝑃2 , 𝛱2 ; [𝑎, 𝑏])| ≤ 𝜀 , where 𝛱𝑖 := 𝛱𝑖󸀠 ∪ 𝛱󸀠󸀠 (𝑖 = 1, 2). From this, we conclude that 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑐]). The assertion 𝑓 ∈ 𝑅𝑆𝛼 ([𝑐, 𝑏]) is proved analogously by considering partitions 𝑃󸀠 ∈ P([𝑎, 𝑐]) and 𝑃1󸀠󸀠 , 𝑃2󸀠󸀠 ∈ P([𝑐, 𝑏]). It remains to prove (4.23). However, any pair of parti­ tions 𝑃󸀠 ∈ P([𝑎, 𝑐]) and 𝑃󸀠󸀠 ∈ P([𝑐, 𝑏]) gives rise to a partition 𝑃 := 𝑃󸀠 ∪ 𝑃󸀠󸀠 ∈ P([𝑎, 𝑏]) which satisfies 𝜇(𝑃) = max {𝜇(𝑃󸀠 ), 𝜇(𝑃󸀠󸀠 )}. For the corresponding RS-sums (involving sets 𝛱󸀠 , 𝛱󸀠󸀠 , and 𝛱 := 𝛱󸀠 ∪ 𝛱󸀠󸀠 ) and RS-integrals, we then get 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) = 𝑆𝛼 (𝑓, 𝑃󸀠 , 𝛱󸀠 ; [𝑎, 𝑐]) + 𝑆𝛼 (𝑓, 𝑃󸀠󸀠 , 𝛱󸀠󸀠 ; [𝑐, 𝑏]) and, passing to the limit 𝜇(𝑃) → 0, 𝑏

𝑐

𝑏

∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = ∫ 𝑓(𝑥) 𝑑𝛼(𝑥) + ∫ 𝑓(𝑥) 𝑑𝛼(𝑥) , 𝑎

𝑎

𝑐

and so we are done. 5 To be precise, we have to assume here that 𝛼 is monotonically increasing because Corollary 4.6 holds only for such 𝛼. However, for 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), the assertion may be obtained by means of the usual Jordan decomposition.

4.1 Classical RS-integrals

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275

The reader might ask why Proposition 4.7 is formulated only in one direction. Recall that for the classical Riemann integral, the other direction is also true: 𝑓 ∈ 𝑅𝑆([𝑎, 𝑐]) and 𝑓 ∈ 𝑅𝑆([𝑐, 𝑏]) implies 𝑓 ∈ 𝑅𝑆([𝑎, 𝑏]) with⁶ 𝑏

𝑐

𝑏

∫ 𝑓(𝑥) 𝑑𝑥 = ∫ 𝑓(𝑥) 𝑑𝑥 + ∫ 𝑓(𝑥) 𝑑𝑥 . 𝑎

𝑎

𝑐

The reason for the missing analogue in Proposition 4.7 is simply that it is not true, as the following example shows. Example 4.8. Let 𝑓 : [−1, 1] → ℝ and 𝛼 : [−1, 1] → ℝ be defined by 𝑓 := 𝜒(0,1] and 𝛼 := 𝜒[−1,0] , i.e. {0 𝑓(𝑥) = { 1 {

{1 for − 1 ≤ 𝑥 ≤ 0 , 𝛼(𝑥) = { 0 for 0 < 𝑥 ≤ 1 . {

for − 1 ≤ 𝑥 ≤ 0 , for 0 < 𝑥 ≤ 1 ,

Clearly, 𝑓 ∈ 𝐵([−1, 1]) and 𝛼 ∈ 𝐵𝑉([−1, 1]). A straightforward calculation shows that 𝑓 ∈ 𝑅𝑆𝛼 ([−1, 0]) ∩ 𝑅𝑆𝛼 ([0, 1]) with 0

1

∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = ∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = 0 . −1

(4.24)

0

On the other hand, both functions 𝑓 and 𝛼 are discontinuous at 0; as we shall later see (Theorem 4.14), this implies that 𝑓 ∈ ̸ 𝑅𝑆𝛼 ([−1, 1]). ♥ The following is a refinement of Example 4.8, insofar as the function 𝑓 is even un­ bounded near its (unique) point of discontinuity. Example 4.9. Let 𝑓 : [−1, 1] → ℝ and 𝛼 : [−1, 1] → ℝ be defined by {0 𝑓(𝑥) := { 1 {𝑥

for − 1 ≤ 𝑥 ≤ 0 , for 0 < 𝑥 ≤ 1 ,

{𝑥 for − 1 ≤ 𝑥 ≤ 0 , 𝛼(𝑥) := { 0 for 0 < 𝑥 ≤ 1 . {

Observe that we even have 𝛼 ∈ 𝐵𝑉([−1, 1])∩𝐶([−1, 1]) in this example. As before, a straightforward calculation shows that 𝑓 ∈ 𝑅𝑆𝛼 ([−1, 0])∩𝑅𝑆𝛼 ([0, 1]) with (4.24). On the other hand, for any partition 𝑃 ∈ P([−1, 1]) which does not contain 0, the upper sum 𝑈𝛼 (𝑓, 𝑃; [−1, 1]) does not exist since 𝑓 is unbounded on (0, 1], and so 𝑓 ∈ ̸ 𝑅𝑆𝛼 ([−1, 1]). ♥ The following proposition gives an important criterion for the existence of Riemann– Stieltjes integrals.

6 Here, the point 𝑐 may also lie outside the interval [𝑎, 𝑏] if we adopt the usual convention that the Riemann integral over [𝑏, 𝑎] is the negative of the Riemann integral over [𝑎, 𝑏].

276 | 4 Riemann–Stieltjes integrals Proposition 4.10. Let 𝑓 ∈ 𝐵([𝑎, 𝑏]), and let 𝛼 : [𝑎, 𝑏] → ℝ be monotonically increasing. Then 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) if and only if for each 𝜀 > 0, there exists 𝛿 > 0 such that (4.5) holds for arbitrary partitions 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) with 𝜇(𝑃) ≤ 𝛿. Proof. Assume first that 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]), let 𝜀 > 0, and choose 𝛿 > 0 such that the as­ sertion of Corollary 4.6 holds for this 𝛿. Fix 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) with 𝜇(𝑃) ≤ 𝛿 1 2 } and 𝛱2 = {𝜏12 , 𝜏22 , . . . , 𝜏𝑚 } of intermediate points as well as two sets 𝛱1 = {𝜏11 , 𝜏21 , . . . , 𝜏𝑚 corresponding to 𝑃1 = 𝑃2 = 𝑃. Applying Corollary 4.6, we then obtain 󵄨󵄨 𝑚 󵄨󵄨 𝑚 󵄨󵄨 󵄨 󵄨󵄨 ∑ 𝑓(𝜏1 )(𝛼(𝑡 ) − 𝛼(𝑡 )) − ∑ 𝑓(𝜏2 )(𝛼(𝑡 ) − 𝛼(𝑡 ))󵄨󵄨󵄨 𝑗 𝑗−1 𝑗 𝑗−1 󵄨󵄨 𝑗 𝑗 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨𝑗=1 𝑗=1 󵄨 𝜀 ≤ |𝑆𝛼 (𝑓, 𝑃, 𝛱1 ; [𝑎, 𝑏]) − 𝑆𝛼 (𝑓, 𝑃, 𝛱2 ; [𝑎, 𝑏])| ≤ . 2

(4.25)

For each 𝑖 ∈ {1, 2, . . . , 𝑚}, we choose the points 𝜏𝑖1 , 𝜏𝑖2 ∈ [𝑡𝑖−1 , 𝑡𝑖 ] in such a way that |𝑓(𝜏𝑖2 ) − 𝑓(𝜏𝑖1 )| = 𝑓(𝜏𝑖2 ) − 𝑓(𝜏𝑖1 ) ≥ 𝑀𝑖 − 𝑚𝑖 −

𝜀 , 2𝑚(𝛼(𝑏) − 𝛼(𝑎))

where 𝑀𝑖 and 𝑚𝑖 are defined as in (4.1) and we may assume without loss of generality that 𝛼(𝑎) < 𝛼(𝑏). Then from (4.25), we obtain 𝑚

𝑈𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) − 𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) = ∑(𝑀𝑖 − 𝑚𝑖 )(𝛼(𝑡𝑖 ) − 𝛼(𝑡𝑖−1 )) 𝑖=1

𝑚

≤ ∑ (𝑓(𝜏𝑖2 ) − 𝑓(𝜏𝑖1 ) + 𝑖=1 𝑚

𝜀 ) (𝛼(𝑡𝑖 ) − 𝛼(𝑡𝑖−1 )) 2𝑚(𝛼(𝑏) − 𝛼(𝑎))

= ∑ (𝑓(𝜏𝑖2 ) − 𝑓(𝜏𝑖1 )) (𝛼(𝑡𝑖 ) − 𝛼(𝑡𝑖−1 )) + 𝑖=1

𝜀 𝑚 𝛼(𝑡𝑖 ) − 𝛼(𝑡𝑖−1 ) ≤ 𝜀. ∑ 2 𝑖=1 𝑚(𝛼(𝑏) − 𝛼(𝑎))

Thus, we have verified that (4.5) holds for every 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]). The converse im­ plication is evident. The following theorem describes two important classes of RS-integrable functions and is analogous to a well-known result for the Riemann integral. Theorem 4.11. (a) For 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), every continuous function 𝑓 : [𝑎, 𝑏] → ℝ is RS-integrable with respect to 𝛼. (b) For 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]), every function 𝑓 : [𝑎, 𝑏] → ℝ of bounded variation is RS-integrable with respect to 𝛼, and the equality (4.20) holds. Proof. By what we have observed above, we may assume without loss of generality that 𝛼 is increasing and nonconstant. (a) Being continuous on the compact interval [𝑎, 𝑏], the function 𝑓 is uniformly continuous. So, given 𝜀 > 0, we find 𝛿 > 0 such that |𝑥 − 𝑦| ≤ 𝛿 implies |𝑓(𝑥) − 𝑓(𝑦)| ≤ 𝜀 for all 𝑥, 𝑦 ∈ [𝑎, 𝑏].

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277

Let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) be any partition satisfying 𝜇(𝑃) ≤ 𝛿, and let 𝑀𝑗 and 𝑚𝑗 be defined by (4.1). Then 𝑚

𝑈𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) − 𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) ≤ (𝛼(𝑏) − 𝛼(𝑎)) ∑(𝑀𝑗 − 𝑚𝑗 ) ≤ (𝛼(𝑏) − 𝛼(𝑎))𝜀 𝑗=1

which shows that 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]). (b) For each 𝑚 ∈ ℕ, we consider a partition 𝑃𝑚 := {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } with the property that 𝛼(𝑏) − 𝛼(𝑎) 𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 ) = (𝑗 = 1, 2, . . . , 𝑚) 𝑚 which is possible by the continuity of 𝛼. Suppose that 𝑓 is monotonically increasing. Then we have 𝑀𝑗 = 𝑓(𝑡𝑗 ) and 𝑚𝑗 = 𝑓(𝑡𝑗−1 ) in (4.1), and hence 𝑈𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) − 𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) =

𝛼(𝑏) − 𝛼(𝑎) 𝑚 ∑ [𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )] 𝑚 𝑗=1

(4.26)

𝛼(𝑏) − 𝛼(𝑎) = [𝑓(𝑏) − 𝑓(𝑎)] ≤ 𝜀 𝑚 if we choose 𝑚 sufficiently large, which shows that 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]). Now, since 𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) ≤ 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) ≤ 𝑈𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) for every partition 𝑃 ∈ P([𝑎, 𝑏]), the estimate (4.26) shows that (4.20) is also true. For general 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), we use again a decomposition argument. Theorem 4.11 contains the well-known result that both continuous and monotone functions are Riemann integrable. Exercise 4.20 shows that the equality (4.20) may fail if we drop the continuity assumption on 𝛼 in Theorem 4.11 (b). We point out that RS-integrals may also be defined for much more general func­ tions than 𝑓 continuous and 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), or 𝑓 monotone and 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]). Both functions 𝑓 and 𝛼 may even be unbounded; however, in this case, they must “interact” in a certain sense in order to “compensate” the unboundedness. Here is an example. Example 4.12. Let 𝑓 : [0, 3] → ℝ and 𝛼 : [0, 3] → ℝ be defined by { 1 𝑓(𝑥) := { 𝑥−1 0 {

for 0 ≤ 𝑥 < 1 , for 1 ≤ 𝑥 ≤ 3 ,

{0 𝛼(𝑥) := { 1 { 𝑥−2

for 0 ≤ 𝑥 ≤ 2 , for 2 < 𝑥 ≤ 3 .

A straightforward calculation shows that 𝑈𝛼 (𝑓, 𝑃; [0, 3]) = 𝐿 𝛼 (𝑓, 𝑃; [0, 3]) = 0 for all 𝑃 ∈ P([0, 3]) satisfying 𝜇(𝑃) < 1/3, and so 𝑓 ∈ 𝑅𝑆𝛼 ([0, 3]). ♥ In view of Theorem 4.11 and Example 4.12, one might ask if (and how) discontinuities of 𝑓 could be “compensated” by 𝛼, or vice versa, to ensure that 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]). For ex­ ample, one may show that “extremely nonintegrable” functions like the Dirichlet func­

278 | 4 Riemann–Stieltjes integrals tion become RS-integrable only for an extremely regular choice of 𝛼 (Exercise 4.17). Likewise, the following proposition shows that a discontinuity of 𝛼 at some point must be “compensated” by the continuity of 𝑓 at this point. For the proof, we introduce some notation. Definition 4.13. Let 𝑀 ⊆ ℝ, 𝑐 ∈ 𝑀, and 𝑓 be a function which is defined on some neighborhood of 𝑐. Then we call the limit osc(𝑓; 𝑐) := lim osc(𝑓; [𝑐 − 𝛿, 𝑐 + 𝛿]) , 𝛿→0+

(4.27)

where osc(𝑓; 𝐴) denotes the oscillation (1.12), the local oscillation of 𝑓 at 𝑐.

◼

Observe that the condition (4.5) for a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) may be written in the form 𝑚

∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ])(𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 ) ≤ 𝜀 .

(4.28)

𝑗=1

This condition will be crucial in the following Theorem 4.14. Let 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) for some 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), and suppose that 𝛼 is dis­ continuous at some point 𝑐 ∈ (𝑎, 𝑏). Then 𝑓 must be continuous at 𝑐. Proof. Suppose that both 𝑓 and 𝛼 are discontinuous at 𝑐. This means, in particular, that osc(𝑓; 𝑐) > 0, where osc(𝑓; 𝑐) > 0 denotes the local oscillation (4.27) of 𝑓 at 𝑐. Concerning the function 𝛼, we distinguish different kinds of discontinuity. Suppose first that 𝛼 has a discontinuity of first kind (jump) at 𝑐, which means that 𝛼(𝑐−) ≠ 𝛼(𝑐+), or a discontinuity of a second kind at 𝑐, which means that 𝛼(𝑐−) or 𝛼(𝑐+) does not exist. Then we find 𝜂 > 0 such that for any 𝛿 > 0, there exist points 𝜎 ∈ [𝑎, 𝑐) and 𝜏 ∈ (𝑐, 𝑏] with 0 < 𝜏 − 𝜎 < 𝛿 and |𝛼(𝜏) − 𝛼(𝜎)| ≥ 𝜂. Now, we fix a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) with the property that 𝜇(𝑃) < 𝛿 and 𝑡𝑖−1 = 𝜎 and 𝑡𝑖 = 𝜏 for some 𝑖 ∈ {1, 2, . . . , 𝑚}. Furthermore, we choose sets 𝛯 := {𝜉1 , 𝜉2 , . . . , 𝜉𝑚 } and 𝐻 := {𝜂1 , 𝜂2 , . . . , 𝜂𝑚 } of points in [𝑎, 𝑏] which satisfy 𝜉𝑗 = 𝜂𝑗 ∈ [𝑡𝑗−1 , 𝑡𝑗 ] for 𝑗 ≠ 𝑖 and 𝜉𝑖 , 𝜂𝑖 ∈ [𝑡𝑖−1 , 𝑡𝑖 ] = [𝜎, 𝜏]. In addition, we may require that 1 |𝑓(𝜉𝑖 ) − 𝑓(𝜂𝑖 )| ≥ osc(𝑓; 𝑐) 2 with osc(𝑓; 𝑐) as above. By construction, we have 𝜂 󵄨 󵄨󵄨 󵄨󵄨𝑆𝛼 (𝑓, 𝑃, 𝛯; [𝑎, 𝑏]) − 𝑆𝛼 (𝑓, 𝑃, 𝐻; [𝑎, 𝑏])󵄨󵄨󵄨 = |𝛼(𝜏) − 𝛼(𝜎)| |𝑓(𝜉𝑖 ) − 𝑓(𝜂𝑖 )| ≥ osc(𝑓; 𝑐) > 0 2 for the corresponding Riemann–Stieltjes sums. Since 𝛿 > 0 was arbitrary, Corollary 4.6 implies that 𝑓 ∈ ̸ 𝑅𝑆𝛼 ([𝑎, 𝑏]), contradicting our hypothesis. Suppose now that 𝛼 has a removable discontinuity at 𝑐, which means that 𝛼(𝑐−) = 𝛼(𝑐+) ≠ 𝛼(𝑐). Then we repeat the above reasoning, but now take 𝜎 := 𝑐 if 𝑓 is right continuous at 𝑐, or 𝜏 := 𝑐 if 𝑓 is left continuous at 𝑐. We then get the same contradiction as before, and the proof is complete.

4.1 Classical RS-integrals

| 279

Observe that the above reasoning is “symmetric” in 𝑓 and 𝛼. This leads to a refined version of Theorem 4.14 which reads as follows. Theorem 4.15. Let 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) for some 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]). Then, at each point of the interval [𝑎, 𝑏], at least one of the functions 𝑓 and 𝛼 is continuous. Proposition 4.3 (a) and (c) and equality (4.15) show that the set 𝑅𝑆𝛼 ([𝑎, 𝑏]) is a linear space. In the following Proposition 4.16, we show that 𝑅𝑆𝛼 ([𝑎, 𝑏]) is also an algebra and prove some other useful properties. Proposition 4.16. Let 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]). Then the following is true. (a) If 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) with 𝑓([𝑎, 𝑏]) ⊆ [𝑐, 𝑑] and ℎ ∈ 𝐶([𝑐, 𝑑]), then ℎ ∘ 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]). (b) If 𝑓, 𝑔 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]), then 𝑓𝑔 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]). (c) If 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]), then |𝑓| ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]). Proof. (a) Let 𝜀 > 0. Since ℎ is uniformly continuous on [𝑐, 𝑑], we may find 𝛿 ∈ (0, 𝜀) such that |ℎ(𝑢) − ℎ(𝑣)| ≤ 𝜀 for all 𝑢, 𝑣 ∈ [𝑐, 𝑑] satisfying |𝑢 − 𝑣| ≤ 𝛿. By assumption, we find a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) such that 𝑈𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) − 𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) ≤ 𝛿2 .

(4.29)

We denote by 𝑀𝑗 and 𝑚𝑗 the same numbers as in (4.1), and by 𝐾𝑗 and 𝑘𝑗 the cor­ responding numbers for ℎ ∘ 𝑓, i.e. 𝐾𝑗 := sup {ℎ(𝑓(𝑥)) : 𝑡𝑗−1 ≤ 𝑥 ≤ 𝑡𝑗 }

(4.30)

𝑘𝑗 := inf {ℎ(𝑓(𝑥)) : 𝑡𝑗−1 ≤ 𝑥 ≤ 𝑡𝑗 } .

(4.31)

and We split the index set {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } into two disjoint parts 𝐼 and 𝐽 by requiring that 𝑗 ∈ 𝐼 if 𝑀𝑗 − 𝑚𝑗 ≤ 𝛿 and 𝑗 ∈ 𝐽 if 𝑀𝑗 − 𝑚𝑗 > 𝛿. For 𝑗 ∈ 𝐼, we have 𝐾𝑗 − 𝑘𝑗 ≤ 𝜀, by our choice of 𝛿, and so ∑(𝐾𝑗 − 𝑘𝑗 )(𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )) ≤ 𝜀 ∑(𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )) ≤ 𝜀 Var(𝛼; [𝑎, 𝑏]) . 𝑗∈𝐼

(4.32)

𝑗∈𝐼

On the other hand, for 𝑗 ∈ 𝐽, we have 𝐾𝑗 − 𝑘𝑗 ≤ 2‖ℎ‖𝐶 . However, (4.29) implies that 𝛿 ∑(𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )) < ∑(𝑀𝑗 − 𝑚𝑗 )(𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )) 𝑗∈𝐽

𝑗∈𝐽

≤ 𝑈𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) − 𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) ≤ 𝛿2 , and hence ∑(𝐾𝑗 − 𝑘𝑗 )(𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )) ≤ 2‖ℎ‖𝐶 ∑(𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 ))

𝑗∈𝐽

𝑗∈𝐽

≤ 2‖ℎ‖𝐶 𝛿 ≤ 2‖ℎ‖𝐶 𝜀 .

(4.33)

280 | 4 Riemann–Stieltjes integrals Combining (4.32) and (4.33), we conclude that 𝑈𝛼 (ℎ ∘ 𝑓, 𝑃; [𝑎, 𝑏]) − 𝐿 𝛼 (ℎ ∘ 𝑓, 𝑃; [𝑎, 𝑏]) ≤ [Var(𝛼; [𝑎, 𝑏]) + 2‖ℎ‖𝐶 ] 𝜀 , and so ℎ ∘ 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) as claimed. (b) Observing that 1 𝑓𝑔 = [(𝑓 + 𝑔)2 − (𝑓 − 𝑔)2 ] , 4 choosing ℎ(𝑢) := 𝑢2 , and using Proposition 4.3 (a) and (c), the assertion follows. (c) The statement follows from (a) for the choice ℎ(𝑢) = |𝑢|. Again, Proposition 4.16 contains a well-known result for the Riemann integral in case 𝛼(𝑥) = 𝑥. The next important theorem shows that in case 𝛼 ∈ 𝐶1 ([𝑎, 𝑏]) the RS-integral (4.7) may always be reduced to the familiar Riemann integral. Theorem 4.17. Let 𝛼 ∈ 𝐶1 ([𝑎, 𝑏]) and 𝑓 ∈ 𝑅𝑆([𝑎, 𝑏]). Then 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) and 𝑏

𝑏

̇ 𝑑𝑡 , ∫ 𝑓(𝑡) 𝑑𝛼(𝑡) = ∫ 𝑓(𝑡)𝛼(𝑡) 𝑎

(4.34)

𝑎

where 𝛼̇ denotes the derivative of 𝛼 with respect to 𝑡, and the integral on the right-hand side of (4.34) is the usual Riemann integral. Proof. Observe first that 𝑓𝛼̇ ∈ 𝑅𝑆([𝑎, 𝑏]) by the well-known properties of the Riemann integral. Let 𝜀 > 0. Then we may find 𝛿1 > 0 such that 󵄨󵄨 󵄨󵄨 𝑚 𝑏 󵄨󵄨 󵄨󵄨 󵄨󵄨 ̇ 𝑗 ) (𝑡𝑗 − 𝑡𝑗−1 ) − ∫ 𝑓(𝑥)𝛼(𝑥) ̇ 𝑑𝑥󵄨󵄨󵄨󵄨 ≤ 𝜀 󵄨󵄨∑ 𝑓(𝜏𝑗 )𝛼(𝜏 󵄨󵄨 󵄨󵄨𝑗=1 𝑎 󵄨󵄨 󵄨󵄨

(4.35)

for any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) with 𝜇(𝑃) ≤ 𝛿1 and any choice of inter­ mediate points 𝜏𝑗 ∈ [𝑡𝑗−1 , 𝑡𝑗 ]. Likewise, we may find 𝛿2 > 0 such that 󵄨󵄨 󵄨󵄨 𝑚 𝑏 󵄨󵄨 󵄨󵄨 󵄨󵄨 ̇ 𝑗 ) (𝑡𝑗 − 𝑡𝑗−1 ) − ∫ 𝛼(𝑥) ̇ 𝑑𝑥󵄨󵄨󵄨󵄨 ≤ 𝜀 󵄨󵄨∑ 𝛼(𝜏 󵄨󵄨 󵄨󵄨𝑗=1 𝑎 󵄨󵄨 󵄨󵄨 for any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) with 𝜇(𝑃) ≤ 𝛿2 and any choice of inter­ mediate points 𝜏𝑗 ∈ [𝑡𝑗−1 , 𝑡𝑗 ]. If 𝜎𝑗 ∈ [𝑡𝑗−1 , 𝑡𝑗 ] is another intermediate point, we have 𝑚

̇ 𝑗 ) − 𝛼(𝜎 ̇ 𝑗 )| (𝑡𝑗 − 𝑡𝑗−1 ) ≤ 2𝜀 ∑ |𝛼(𝜏

(4.36)

𝑗=1

whenever 𝜇(𝑃) ≤ 𝛿2 because 𝛼̇ is continuous. Now, we fix any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) satisfying 𝜇(𝑃) ≤ 𝛿 := min {𝛿1 , 𝛿2 }. By the mean value theorem, we find points 𝜎𝑗 ∈ [𝑡𝑗−1 , 𝑡𝑗 ] such that ̇ 𝑗 ) (𝑡𝑗 − 𝑡𝑗−1 ) . 𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 ) = 𝛼(𝜎

4.1 Classical RS-integrals

| 281

Consequently, 𝑚

∑ 𝑓(𝜏𝑗 ) [𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )]

𝑗=1

𝑚

(4.37)

𝑚

̇ 𝑗 ) (𝑡𝑗 − 𝑡𝑗−1 ) + ∑ 𝑓(𝜏𝑗 ) [𝛼(𝜏 ̇ 𝑗 ) − 𝛼(𝜎 ̇ 𝑗 )] (𝑡𝑗 − 𝑡𝑗−1 ) . = ∑ 𝑓(𝜏𝑗 )𝛼(𝜎 𝑗=1

𝑗=1

However, (4.35) and (4.36) imply that 󵄨󵄨 󵄨󵄨 𝑚 𝑏 󵄨󵄨 󵄨󵄨 󵄨󵄨 ̇ 𝑑𝑥󵄨󵄨󵄨󵄨 ≤ (2‖𝑓‖∞ + 1)𝜀. 󵄨󵄨∑ 𝑓(𝜏𝑗 ) [𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )] − ∫ 𝑓(𝑥)𝛼(𝑥) 󵄨󵄨 󵄨󵄨𝑗=1 𝑎 󵄨󵄨 󵄨󵄨 This shows that 𝑏

𝑚

̇ 𝑑𝑥, lim ∑ 𝑓(𝜏𝑗 ) [𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )] = lim 𝑆𝛼̇ (𝑓, 𝑃; [𝑎, 𝑏]) = ∫ 𝑓(𝑥)𝛼(𝑥)

𝜇(𝑃)→0

𝜇(𝑃)→0

𝑗=1

𝑎

by Proposition 4.5, which proves the assertion. An extension of Theorem 4.17 from functions 𝛼 ∈ 𝐶1 ([𝑎, 𝑏]) to functions 𝛼 ∈ 𝐴𝐶([𝑎, 𝑏]) is given in Theorem 4.43 in Section 4.5. In Proposition 4.10, we have given a necessary and sufficient condition for the RSintegrability of a function 𝑓 with respect to some increasing integrator 𝛼. The follow­ ing theorem from [182] gives a more complete picture and is valid for general functions 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]). Theorem 4.18. Let 𝑓 ∈ 𝐵([𝑎, 𝑏]) and 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]). Then the following three statements are equivalent. (a) 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]). (b) For each 𝜀 > 0, there exists 𝛿 > 0 such that for any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) with 𝜇(𝑃) ≤ 𝛿, we have 𝑚

∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ])|𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )| ≤ 𝜀,

(4.38)

𝑗=1

where osc(𝑓; 𝐴) denotes the oscillation (1.12) of 𝑓 on 𝐴 ⊆ [𝑎, 𝑏]. (c) For each 𝜀 > 0, there exists 𝛿 > 0 such that for any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) with 𝜇(𝑃) ≤ 𝛿, we have 𝑚

∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) Var(𝛼; [𝑡𝑗−1 , 𝑡𝑗 ]) ≤ 𝜀.

(4.39)

𝑗=1

Proof. We show first that (a) implies (b). Let 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]). Since 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), ac­ cording to Theorem 1.5, we can represent 𝛼 as the difference 𝛼 = 𝛽 − 𝛾 of two monoton­ ically increasing functions 𝛽, 𝛾 : [𝑎, 𝑏] → ℝ, and so (4.15) holds. Given 𝜀 > 0, by Propo­ sition 4.10, we can find 𝛿 > 0 such that for an arbitrary partition 𝑃 = {𝑡1 , 𝑡2 , . . . , 𝑡𝑚 } ∈

282 | 4 Riemann–Stieltjes integrals P([𝑎, 𝑏]) with 𝜇(𝑃) ≤ 𝛿, we have 𝑚

∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ])(𝛽(𝑡𝑗 ) − 𝛽(𝑡𝑗−1 )) ≤

𝑗=1

and

𝑚

∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ])(𝛾(𝑡𝑗 ) − 𝛾(𝑡𝑗−1 )) ≤

𝑗=1

𝜀 2 𝜀 . 2

Keeping in mind the above estimates, we get 𝑚

∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ])|𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )|

𝑗=1

𝑚

= ∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ])|𝛽(𝑡𝑗 ) − 𝛾(𝑡𝑗 ) − 𝛽(𝑡𝑗−1 ) + 𝛾(𝑡𝑗−1 )| 𝑗=1 𝑚

= ∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ])|[𝛽(𝑡𝑗 ) − 𝛽(𝑡𝑗−1 )] − [𝛾(𝑡𝑗 ) − 𝛾(𝑡𝑗−1 )]| 𝑗=1 𝑚

≤ ∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ])[|𝛽(𝑡𝑗 ) − 𝛽(𝑡𝑗−1 )| + |𝛾(𝑡𝑗 ) − 𝛾(𝑡𝑗−1 )|] 𝑗=1 𝑚

= ∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ])(𝛽(𝑡𝑗 ) − 𝛽(𝑡𝑗−1 )) 𝑗=1

𝑚

+ ∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ])(𝛾(𝑡𝑗 ) − 𝛾(𝑡𝑗−1 )) ≤ 𝜀 . 𝑗=1

This shows that condition (b) holds, and so we have proved the first implication. Now, we prove that (b) implies (c). We first show⁷ that under condition (b), for each point 𝑥 ∈ [𝑎, 𝑏], at least one of the functions 𝑓 or 𝛼 is continuous at 𝑥. Given 𝜀 > 0, choose 𝛿 > 0 according to condition (b). Fix a point 𝑥0 ∈ [𝑎, 𝑏] and assume, for example, that 𝛼 is discontinuous at 𝑥0 . Then we have lim 𝛼(𝑥) ≠ 𝛼(𝑥0 )

𝑥→𝑥0+

or lim 𝛼(𝑥) ≠ lim 𝛼(𝑥) .

𝑥→𝑥0−

𝑥→𝑥0+

Further, take a number ℎ > 0 such that ℎ < min{𝛿, 𝑏 − 𝑥0 } and consider an arbitrary partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) with 𝜇(𝑃) ≤ 𝛿. We add to the partition 𝑃 the two points 𝑥ℎ and 𝑥0 + ℎ, where 𝑥ℎ = 𝑥0 or 𝑥ℎ = 𝑥0 − ℎ, and denote by 𝑃󸀠 the extended partition 𝑃 ∪ {𝑥ℎ , 𝑥0 + ℎ}. Obviously, 𝜇(𝑃󸀠 ) ≤ 𝛿 and, in view of (4.38), we have |𝑓(𝑥0 + ℎ) − 𝑓(𝑥0 )||𝛼(𝑥0 + ℎ) − 𝛼(𝑥ℎ )| ≤ 𝜀 .

7 Note that we cannot apply Theorem 4.14 here since we do not suppose that 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]).

4.1 Classical RS-integrals

| 283

Hence, letting ℎ → 0, we get 𝑓(𝑥0 + ℎ) → 𝑓(𝑥0 ) which means that 𝑓 is right contin­ uous at 𝑥0 . Similarly, we can prove the left continuity of 𝑓 at 𝑥0 , and so we conclude that the function 𝑓 is continuous at 𝑥0 . Repeating the above reasoning under the assumption that the function 𝑓 is dis­ continuous at some point 𝑥0 , we may prove that the function 𝛼 is continuous at 𝑥0 . So, we proved the assertion. Fix 𝜀 > 0. Choose 𝛿0 > 0 according to condition (b) corresponding to the num­ ber 𝜀/3. Let 𝑀 := osc(𝑓; [𝑎, 𝑏]), and let 𝑃 = {𝑧0 , 𝑧1 , . . . , 𝑧𝑘 } ∈ P([𝑎, 𝑏]) be a partition satisfying⁸ 𝑘 𝜀 ≤ ∑ |𝛼(𝑧𝑗 ) − 𝛼(𝑧𝑗−1 )|. (4.40) Var(𝛼, [𝑎, 𝑏]) − 3𝑀 𝑗=1 Obviously, if we take any partition 𝑃󸀠 ⊇ 𝑃, then (4.40) also holds for 𝑃󸀠 . As we have seen, condition (b) implies that at every point of the interval [𝑎, 𝑏], at least one of the functions 𝑓 or 𝛼 is continuous. In view of Proposition 1.7, this implies that the same is true for at least one of the functions 𝑓 or 𝑉𝛼 , where 𝑉𝛼 denotes the variation function (1.13) of 𝛼. Since these functions are bounded, we can find 𝛿1 > 0 such that 𝜀 (4.41) osc(𝑓; [𝑥󸀠 , 𝑥]) Var(𝛼; [𝑥󸀠 , 𝑥]) ≤ 3𝑀 provided that |𝑥 − 𝑥󸀠 | ≤ 𝛿1 and 𝑥󸀠 < 𝑧𝑗 < 𝑥 for 𝑗 = 1, 2, . . . , 𝑘. Next, let 𝑃1 = {𝑥0 , 𝑥1 , . . . , 𝑥𝑛 } ∈ P([𝑎, 𝑏]) be such that 𝜇(𝑃1 ) ≤ 𝛿 := min {𝛿0 , 𝛿1 }. Obviously, inequality (4.41) holds for the partition 𝑃1 with 𝜀 replaced by 𝜀/3. Extending the partition 𝑃1 by including those points 𝑧𝑗 which are distinct from points of 𝑃1 , we obtain a new partition 𝑃2 = {𝑥󸀠0 , 𝑥󸀠1 , . . . , 𝑥󸀠𝑝 } satisfying 𝑝

Var(𝛼; [𝑎, 𝑏]) −

𝜀 ≤ ∑ |𝛼(𝑥󸀠𝑗 ) − 𝛼(𝑥󸀠𝑗−1 )| . 3𝑀 𝑗=1

Consequently, we get 𝑝

∑ [Var(𝛼; [𝑥󸀠𝑗−1 , 𝑥󸀠𝑗 ]) − |𝛼(𝑥󸀠𝑗 ) − 𝛼(𝑥󸀠𝑗−1 )|] ≤

𝑗=1

𝜀 . 3𝑀

This implies 𝑝

𝑆0 := ∑ osc(𝑓; [𝑥󸀠𝑗−1 , 𝑥󸀠𝑗 ]) Var(𝛼; [𝑥󸀠𝑗−1 , 𝑥󸀠𝑗 ]) 𝑗=1 𝑝

≤ ∑ osc(𝑓; [𝑥󸀠𝑗−1 , 𝑥󸀠𝑗 ]){Var(𝛼; [𝑥󸀠𝑗−1 , 𝑥󸀠𝑗 ]) − |𝛼(𝑥󸀠𝑗 ) − 𝛼(𝑥󸀠𝑗−1 )|} 𝑗=1

𝑝

+ ∑ osc(𝑓; [𝑥󸀠𝑗−1 , 𝑥󸀠𝑗 ])|𝛼(𝑥󸀠𝑗 ) − 𝛼(𝑥󸀠𝑗−1 )| ≤ 𝑗=1

2 𝜀. 3

8 Of course, without loss of generality, we may assume that 𝑓 is not constant.

284 | 4 Riemann–Stieltjes integrals Finally, we notice that we can write 𝑛

𝑆 := ∑ osc(𝑓; [𝑥𝑗−1 , 𝑥𝑗 ]) Var(𝛼; [𝑥𝑗−1 , 𝑥𝑗 ]) = 𝑆󸀠 + 𝑆󸀠󸀠 , 𝑗=1

where 𝑆󸀠󸀠 denotes the sum of those terms for which there exists 𝑘 such that 𝑥𝑗−1 < 𝑧𝑘 < 𝑥𝑗 , while 𝑆󸀠 denotes the remainder sum of 𝑆, being a partial sum of 𝑆0 . Obviously, we have 𝑆󸀠 ≤ 𝑆0 ≤ 2𝜀/3. Furthermore, from (4.41), we know that 𝑆󸀠󸀠 ≤ 𝜀/3, and so 𝑆 ≤ 𝜀 as claimed. It remains to show that (c) implies (a). To this end, we represent the function 𝛼 again in the form 𝛼 = 𝛽 − 𝛾, where⁹ 𝛽(𝑥) =

1 [𝑉 (𝑥) + 𝛼(𝑥)] , 2 𝛼

𝛾(𝑥) =

1 [𝑉 (𝑥) − 𝛼(𝑥)] , 2 𝛼

and 𝑉𝛼 denotes the variation function (1.13) of 𝛼. Clearly , Var(𝛼; [𝑥, 𝑦]) = Var(𝛽; [𝑥, 𝑦]) + Var(𝛾; [𝑥, 𝑦])

(4.42)

for an arbitrary interval [𝑥, 𝑦] ⊆ [𝑎, 𝑏]. Fix 𝜀 > 0 and choose 𝛿 > 0 according to con­ dition (c). Then, in view of (4.39), for any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) with 𝜇(𝑃) ≤ 𝛿, we have 𝑚

∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) Var(𝛼; [𝑡𝑗−1 , 𝑡𝑗 ]) ≤ 𝜀 .

𝑗=1

From (4.42), we then obtain the inequality 𝑚

∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]){Var(𝛽; [𝑡𝑗−1 , 𝑡𝑗 ]) + Var(𝛾; [𝑡𝑗−1 , 𝑡𝑗 ])} ≤ 𝜀 ,

𝑗=1

and hence 𝑚

𝑚

∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ])(𝛽(𝑡𝑗 ) − 𝛽(𝑡𝑗−1 )) = ∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) Var(𝛽; [𝑡𝑗−1 , 𝑡𝑗 ]) ≤ 𝜀

𝑗=1

𝑗=1

𝑚

𝑚

and ∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ])(𝛾(𝑡𝑗 ) − 𝛾(𝑡𝑗−1 )) = ∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) Var(𝛾; [𝑡𝑗−1 , 𝑡𝑗 ]) ≤ 𝜀 .

𝑗=1

𝑗=1

From these estimates and Proposition 4.10, we deduce that both 𝑓 ∈ 𝑅𝑆𝛽 ([𝑎, 𝑏]) and 𝑓 ∈ 𝑅𝑆𝛾 ([𝑎, 𝑏]), and so by the definition, the RS-integral for 𝐵𝑉-functions, that 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]). This completes the proof.

9 This is the Jordan decomposition involving “slowly increasing” functions, see Exercise 1.27.

4.1 Classical RS-integrals

| 285

Observe that the implication (c) ⇒ (b) in Theorem 4.18 is trivial by (1.8). However, the nontrivial implication (b) ⇒ (c) in our proof shows that the difference is not essential if 𝜇(𝑃) is “small enough.” Theorem 4.18 has a useful consequence which relates RS-integrability of a func­ tion 𝑓 with respect to a function 𝛼 and that with respect to its variation function 𝑉𝛼 . Corollary 4.19. Let 𝑓 ∈ 𝐵([𝑎, 𝑏]) and 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]). Then 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) if and only if 𝑓 ∈ 𝑅𝑆𝑉𝛼 ([𝑎, 𝑏]), where 𝑉𝛼 denotes the variation function (1.13) of 𝛼. Proof. Being increasing, the function 𝑉𝛼 has bounded variation on [𝑎, 𝑏]. Now, 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) implies (4.38), and so also (4.39), by Theorem 4.18. However, the equality Var(𝛼; [𝑡𝑗−1 , 𝑡𝑗 ]) = 𝑉𝛼 (𝑡𝑗 ) − 𝑉𝛼 (𝑡𝑗−1 )

(4.43)

shows that then 𝑓 ∈ 𝑅𝑆𝑉𝛼 ([𝑎, 𝑏]), again by Theorem 4.18. Since this argument is sym­ metric in 𝛼 and 𝑉𝛼 , we have proved the desired equivalence. In view of Corollary 4.19, the question arises if there is any relation between the RSintegral of a function 𝑓 with respect to 𝑉𝛼 and its RS-integral with respect to its parent function 𝛼. The following theorem contains such a relation which gives a useful esti­ mate for RS-integrals with interesting applications to integral equations of fractional order [44]. We did not find the proof of this estimate in the literature,¹⁰ and so we give a short proof here. Theorem 4.20. For 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]), the estimate 󵄨󵄨 𝑏 󵄨󵄨 𝑏 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨∫ 𝑓(𝑥) 𝑑𝛼(𝑥)󵄨󵄨󵄨 ≤ ∫ |𝑓(𝑥)| 𝑑𝑉𝛼 (𝑥) 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑎 󵄨󵄨 𝑎

(4.44)

holds. Proof. From Corollary 4.19, we know that 𝑓 ∈ 𝑅𝑆𝑉𝛼 ([𝑎, 𝑏]), and so the integral appear­ ing on the right-hand side of (4.44) is well-defined. Take an arbitrary partition {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), and choose a set of interme­ diate points 𝛱 = {𝜏1 , 𝜏2 , . . . , 𝜏𝑚 } for the partition 𝑃 satisfying (4.16). Then 󵄨󵄨 𝑚 󵄨󵄨 𝑚 󵄨󵄨 󵄨 󵄨󵄨∑ 𝑓(𝜏 )(𝛼(𝑡 ) − 𝛼(𝑡 ))󵄨󵄨󵄨 ≤ ∑ |𝑓(𝜏 )| |𝛼(𝑡 ) − 𝛼(𝑡 )| 󵄨󵄨 𝑗 𝑗 𝑗−1 󵄨󵄨 𝑗 𝑗 𝑗−1 󵄨󵄨𝑗=1 󵄨󵄨 𝑗=1 󵄨 󵄨 𝑚

𝑚

≤ ∑ |𝑓(𝜏𝑗 )| Var(𝛼; [𝑡𝑗−1 , 𝑡𝑗 ]) = ∑ |𝑓(𝜏𝑗 )|(𝑉𝛼 (𝑡𝑗 ) − 𝑉𝛼 (𝑡𝑗−1 )) , 𝑗=1

𝑗=1

where we have used (1.8). Since the above inequality is satisfied for arbitrary partitions 𝑃 ∈ P([𝑎, 𝑏]) and arbitrary sets 𝛱 of intermediate points, in view of Proposition 4.5 and the continuity of the function 𝑢 󳨃→ |𝑢|, we obtain (4.44).

10 In some textbooks and monographs, e.g. [237], this estimate is mentioned without proof.

286 | 4 Riemann–Stieltjes integrals We point out that the estimate (4.44) is not true if we also put 𝑑𝛼(𝑥), rather than 𝑑𝑉𝛼 (𝑥), on the right-hand side, see Exercise 4.42. For another proof of the important Theo­ rem 4.20, see Exercise 4.44. In the next theorem, we prove a fundamental estimate for the RS-integral and deduce two natural convergence theorems. Theorem 4.21. The RS-integral has the following properties. (a) The fundamental estimate 󵄨󵄨 𝑏 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨∫ 𝑓(𝑡) 𝑑𝛼(𝑡)󵄨󵄨󵄨 ≤ sup |𝑓(𝑥)| Var(𝛼; [𝑎, 𝑏]) 󵄨󵄨 𝑎 󵄨󵄨 𝑎≤𝑥≤𝑏 󵄨 󵄨

(4.45)

holds for all 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) and 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]). (b) If (𝑓𝑛 )𝑛 is a sequence of bounded functions 𝑓𝑛 : [𝑎, 𝑏] → ℝ which satisfies ‖𝑓𝑛 − 𝑓‖∞ → 0 as 𝑛 → ∞, then 𝑏

𝑏

lim ∫ 𝑓𝑛 (𝑥) 𝑑𝛼(𝑥) = ∫ 𝑓(𝑥) 𝑑𝛼(𝑥)

𝑛→∞

𝑎

(4.46)

𝑎

for all 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]). (c) If (𝛼𝑛 )𝑛 is a sequence of functions 𝛼𝑛 : [𝑎, 𝑏] → ℝ of bounded variation which satis­ fies Var(𝛼𝑛 − 𝛼; [𝑎, 𝑏]) → 0 as 𝑛 → ∞, then 𝑏

𝑏

lim ∫ 𝑓(𝑥) 𝑑𝛼𝑛 (𝑥) = ∫ 𝑓(𝑥) 𝑑𝛼(𝑥)

𝑛→∞

𝑎

(4.47)

𝑎

for all 𝑓 ∈ 𝐵([𝑎, 𝑏]). Proof. As before, we may assume without loss of generality that 𝛼 is monotonically increasing. By 𝑉𝛼 , we denote the variation function (1.13) of 𝛼. (a) In view of Theorem 4.20 and Proposition 4.3 (e), we obtain 󵄨󵄨 𝑏 󵄨󵄨 𝑏 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨∫ 𝑓(𝑥) 𝑑𝛼(𝑥)󵄨󵄨󵄨 ≤ ∫ |𝑓(𝑥)| 𝑑𝑉𝛼 (𝑥) 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑎 󵄨󵄨 𝑎 𝑏

≤ ‖𝑓‖∞ ∫ 𝑑𝑉𝛼 (𝑥) = sup |𝑓(𝑥)| Var(𝛼; [𝑎, 𝑏]) 𝑎

𝑎≤𝑥≤𝑏

as claimed. (b) Using again Theorem 4.20 and Proposition 4.3 (e), we get 󵄨󵄨 𝑏 󵄨󵄨 𝑏 𝑏 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨∫ 𝑓𝑛 (𝑥) 𝑑𝛼(𝑥) − ∫ 𝑓(𝑥) 𝑑𝛼(𝑥)󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨𝑓𝑛 (𝑥) − 𝑓(𝑥)󵄨󵄨 𝑑𝑉𝛼 (𝑥) ≤ ‖𝑓𝑛 − 𝑓‖∞ Var(𝛼; [𝑎, 𝑏]) , 󵄨 󵄨󵄨 𝑎 󵄨 𝑎 󵄨 𝑎 󵄨 which proves the assertion.

4.1 Classical RS-integrals

| 287

(c) Applying (4.44) with 𝛼 replaced by 𝛼𝑛 − 𝛼, we obtain 󵄨󵄨 󵄨󵄨 𝑏 󵄨󵄨 󵄨󵄨 𝑏 𝑏 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨∫ 𝑓(𝑥) 𝑑𝛼𝑛 (𝑥) − ∫ 𝑓(𝑥) 𝑑𝛼(𝑥)󵄨󵄨 = 󵄨󵄨∫ 𝑓(𝑥) 𝑑(𝛼𝑛 (𝑥) − 𝛼(𝑥))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑎 󵄨󵄨 󵄨󵄨 𝑎 󵄨󵄨 𝑎 𝑏

≤ ∫ |𝑓(𝑥)| 𝑑𝑉𝛼𝑛 −𝛼 (𝑥) ≤ ‖𝑓‖∞ Var(𝛼𝑛 − 𝛼; [𝑎, 𝑏]) , 𝑎

and the assertion follows. Theorem 4.21 shows that the map 𝑏

(𝑓, 𝛼) 󳨃→ ∫ 𝑓(𝑥) 𝑑𝛼(𝑥) 𝑎

is a continuous bilinear form on the product 𝐶([𝑎, 𝑏]) × 𝐵𝑉([𝑎, 𝑏]). This bilinear form will play an important role in the following two sections. Of course, the convergence results contained in Theorem 4.21 (b) and (c) are not too surprising since they refer to the “natural” norms on 𝐵([𝑎, 𝑏]) (uniform conver­ gence) and 𝐵𝑉([𝑎, 𝑏]) (convergence in variation). Recall that the estimate |𝛼𝑛 (𝑥) − 𝛼(𝑥)| ≤ Var(𝛼𝑛 − 𝛼; [𝑎, 𝑏])

(𝑎 ≤ 𝑥 ≤ 𝑏)

implies that under the hypotheses of Theorem 4.21 (c), the sequence (𝛼𝑛 )𝑛 also con­ verges uniformly on [𝑎, 𝑏] to 𝛼. The following example shows that (4.47) may fail if we only assume uniform convergence of (𝛼𝑛 )𝑛 in 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]) on [𝑎, 𝑏] to some function 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]). Example 4.22. Let 𝑓 : [0, 2𝜋] → ℝ and 𝛼𝑛 : [0, 2𝜋] → ℝ be defined by ∞

𝑓(𝑥) := ∑ 𝑚=1

cos 𝑚6 𝑥 , 𝑚2

𝛼𝑛 (𝑥) :=

sin 𝑛𝑥 √𝑛

(𝑛 = 1, 2, 3 . . .) .

(4.48)

Since the series in (4.48) converges uniformly on [0, 2𝜋], the function 𝑓 is contin­ uous; moreover, the sequence (𝛼𝑛 )𝑛 obviously converges uniformly on [0, 2𝜋] to zero. Since 𝛼𝑛 ∈ 𝐶1 ([0, 2𝜋]), we may use Theorem 4.17 to calculate the RS-integral of 𝑓 with respect to 𝛼𝑛 and obtain 2𝜋

2𝜋

2𝜋

𝐼𝑛 := ∫ 𝑓(𝑥) 𝑑𝛼𝑛 (𝑥) = ∫ 𝑓(𝑥)𝛼𝑛̇ (𝑥) 𝑑𝑥 = √𝑛 ∫ 𝑓(𝑥) cos 𝑛𝑥 𝑑𝑥 . 0

0

0

Putting the definition (4.48) of 𝑓 in the integral and observing that 2𝜋

{𝜋 if 𝑚 = 𝑛 , ∫ cos 𝑚𝑥 cos 𝑛𝑥 𝑑𝑥 = { 0 if 𝑚 ≠ 𝑛 , 0 {

288 | 4 Riemann–Stieltjes integrals we end up with 2𝜋

6 6 {√𝑛 ∫0 cos2 𝑛𝑥 𝑑𝑥 = 𝜋√𝑛 if 𝑛 = 𝑝6 for some 𝑝 ∈ ℕ , 𝐼𝑛 = { 0 otherwise . {

This shows that the sequence of integrals (𝐼𝑛 )𝑛 is unbounded, and so it cannot be ♥ convergent, although the sequence (𝛼𝑛 )𝑛 converges uniformly on ℝ to 0. The crucial point in Example 4.22 is of course that we have chosen the “wrong type of convergence” for the sequence (𝛼𝑛 )𝑛 . From Theorem 4.21 (c), it follows that (𝛼𝑛 )𝑛 cannot converge in the norm of 𝐵𝑉([0, 2𝜋]) to 0. Indeed, consider the partition 𝑃𝑛 := {0, 2𝜋} ∪ {𝑠0 , 𝑠1 , . . . , 𝑠𝑛 } ∪ {𝑡0 , 𝑡1 , . . . , 𝑡𝑛 } ∈ P([0, 2𝜋]) with points (3 + 4𝑘)𝜋 (𝑘 = 1, 2, . . . , 𝑛) . 2𝑛 From 𝛼𝑛 (𝑠𝑘 ) = 1/√𝑘 and 𝛼𝑛 (𝑡𝑘 ) = −1/√𝑘, it follows then that 𝑠𝑘 :=

(1 + 4𝑘)𝜋 , 2𝑛

𝑡𝑘 :=

𝑛

2 → ∞ (𝑛 → ∞) , √ 𝑘 𝑘=1

Var(𝛼𝑛 ; [0, 2𝜋]) ≥ Var(𝛼𝑛 , 𝑃𝑛 ; [0, 2𝜋]) ≥ ∑

which shows that the sequence (𝛼𝑛 )𝑛 is unbounded in the norm of 𝐵𝑉([0, 2𝜋]). In some cases, the fundamental estimate (4.45) in Theorem 4.21 may be sharp­ ened. For example, the following estimates are useful in the theory of Fourier series: Proposition 4.23. Let 𝛼 ∈ 𝐵𝑉([0, 2𝜋]). Then the following is true. (a) The estimate 󵄨󵄨 󵄨󵄨 2𝜋 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 ∫ 𝛼(𝑥) cos 𝑛𝑥 𝑑𝑥󵄨󵄨󵄨 ≤ 1 Var(𝛼; [0, 2𝜋]) (4.49) 󵄨󵄨 𝑛 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨0 holds for all 𝑛 ∈ ℕ; moreover, there exists a function 𝛼̃ ∈ 𝐵𝑉([0, 2𝜋]) such that 󵄨󵄨󵄨 󵄨󵄨󵄨 2𝜋 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 = Var(𝛼;̃ [0, 2𝜋]) 󵄨󵄨 ∫ 𝛼(𝑥) ̃ cos 𝑛𝑥 𝑑𝑥 (4.50) 󵄨󵄨 𝑛 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨0 for infinitely many 𝑛 ∈ ℕ. (b) The estimate 󵄨󵄨 2𝜋 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 󵄨 (4.51) 󵄨󵄨 ∫ 𝛼(𝑥) sin 𝑛𝑥 𝑑𝑥󵄨󵄨󵄨 ≤ Var(𝛼; [0, 2𝜋]) 󵄨󵄨 󵄨󵄨 𝑛 󵄨󵄨 0 󵄨󵄨 holds for all 𝑛 ∈ ℕ; moreover, there exists a function 𝛼̂ ∈ 𝐵𝑉([0, 2𝜋]) such that 󵄨󵄨 2𝜋 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 ̂ 󵄨 󵄨󵄨󵄨 ∫ 𝛼(𝑥) sin 𝑛𝑥 𝑑𝑥󵄨󵄨󵄨 = 𝑛 Var(𝛼;̂ [0, 2𝜋]) 󵄨󵄨 󵄨󵄨 󵄨 󵄨0 for infinitely many 𝑛 ∈ ℕ.

(4.52)

4.1 Classical RS-integrals

| 289

Proof. Suppose first that 𝛼 : [0, 2𝜋] → ℝ is increasing. Since 2𝜋

∫ cos 𝑛𝑥 𝑑𝑥 = 0 (𝑛 = 1, 2, 3, . . .) , 0

we may suppose that 𝛼(0) = 0 and so Var(𝛼; [0, 2𝜋]) = 𝛼(2𝜋). From the third mean value theorem for the Riemann integral (Exercise 4.25), we get 󵄨󵄨 2𝜋 󵄨󵄨 󵄨󵄨 󵄨󵄨 2𝜋 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∫ 𝛼(𝑥) cos 𝑛𝑥 𝑑𝑥 ≤ 𝛼(2𝜋) sup ∫ cos 𝑛𝑥 𝑑𝑥 󵄨󵄨 ≤ Var(𝛼; [0, 2𝜋]) . 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 𝑛 󵄨 0≤𝑐≤2𝜋 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑐 󵄨󵄨 󵄨0 󵄨 󵄨

(4.53)

In the general case 𝛼 ∈ 𝐵𝑉([0, 2𝜋]), we may find increasing functions 𝛽 and 𝛾 such that 𝛼 = 𝛽−𝛾 and Var(𝛼; [0, 2𝜋]) = Var(𝛽; [0, 2𝜋])+Var(𝛾; [0, 2𝜋]), see Exercise 1.27. Then we obtain 󵄨󵄨 󵄨󵄨 2𝜋 󵄨󵄨 󵄨󵄨 2𝜋 󵄨󵄨 󵄨󵄨 2𝜋 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 ∫ 𝛼(𝑥) cos 𝑛𝑥 𝑑𝑥󵄨󵄨 ≤ 󵄨󵄨 ∫ 𝛽(𝑥) cos 𝑛𝑥 𝑑𝑥󵄨󵄨 + 󵄨󵄨 ∫ 𝛾(𝑥) cos 𝑛𝑥 𝑑𝑥󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 󵄨󵄨 0 1 1 1 ≤ Var(𝛽; [0, 2𝜋]) + Var(𝛾; [0, 2𝜋]) = Var(𝛼; [0, 2𝜋]) 𝑛 𝑛 𝑛 ̃ as claimed. Choosing, in particular, 𝛼(𝑥) := 𝜒[0,𝜋/2] (𝑥) for 0 ≤ 𝑥 ≤ 2𝜋, we have Var(𝛼;̃ [0, 2𝜋]) = 1 and 󵄨󵄨 󵄨󵄨 2𝜋 󵄨󵄨 󵄨󵄨󵄨 𝜋/2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 1 󵄨󵄨 𝑛𝜋 󵄨󵄨󵄨 1 󵄨󵄨 ̃ 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 󵄨󵄨 ∫ 𝛼(𝑥) cos 𝑛𝑥 𝑑𝑥󵄨󵄨 = 󵄨󵄨 ∫ cos 𝑛𝑥 𝑑𝑥󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨sin 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 𝑛 󵄨󵄨 𝑛 󵄨 󵄨󵄨 󵄨󵄨 0 󵄨󵄨 󵄨󵄨 0 for all odd 𝑛 ∈ ℕ, and so we have proved (a). The proof of (b) is very similar. We start again with an increasing function and replace (4.53) by 󵄨󵄨 󵄨󵄨 󵄨󵄨 2𝜋 󵄨󵄨 2𝜋 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∫ 𝛼(𝑥) sin 𝑛𝑥 𝑑𝑥󵄨󵄨 ≤ 𝛼(2𝜋) sup 󵄨󵄨 ∫ sin 𝑛𝑥 𝑑𝑥󵄨󵄨󵄨 ≤ Var(𝛼; [0, 2𝜋]) . 󵄨󵄨 󵄨󵄨 𝑛 󵄨󵄨 󵄨󵄨 0≤𝑐≤2𝜋 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑐 󵄨󵄨 0

(4.54)

The remaining part using the Jordan decomposition of 𝛼 ∈ 𝐵𝑉([0, 2𝜋]) is precisely ̂ the same as in (a). To prove the last assertion, we choose now 𝛼(𝑥) := 𝜒[0,𝜋] (𝑥) for 0 ≤ 𝑥 ≤ 2𝜋 and get Var(𝛼;̂ [0, 2𝜋]) = 1 and 󵄨󵄨 󵄨󵄨 𝜋 󵄨󵄨 󵄨󵄨 2𝜋 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 2 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ̂ 󵄨󵄨 ∫ 𝛼(𝑥) sin 𝑛𝑥 𝑑𝑥󵄨󵄨 = 󵄨󵄨∫ sin 𝑛𝑥 𝑑𝑥󵄨󵄨󵄨 = |1 − cos 𝑛𝜋| = 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑛 󵄨󵄨 𝑛 󵄨󵄨 󵄨󵄨 0 󵄨󵄨 󵄨󵄨 0 for all odd 𝑛 ∈ ℕ. Now, we present two useful results on RS-integrals which for the Riemann integral (i.e. 𝛼(𝑥) = 𝑥) reduce to well-known integration techniques.

290 | 4 Riemann–Stieltjes integrals Proposition 4.24 (integration by parts). Let 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]) and 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]). Then the equality 𝑏

𝑏

∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = 𝑓(𝑏)𝛼(𝑏) − 𝑓(𝑎)𝛼(𝑎) − ∫ 𝛼(𝑥) 𝑑𝑓(𝑥) 𝑎

(4.55)

𝑎

holds. Proof. Let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) and 𝛱 = {𝜏0 , 𝜏1 , . . . , 𝜏𝑚 , 𝜏𝑚+1 } be a set of inter­ mediate points satisfying 𝜏0 = 𝑎, 𝜏𝑚+1 = 𝑏, and (4.16). For the RS-sums (4.17), we then obtain¹¹ 𝑚

𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) = ∑ 𝑓(𝜏𝑗 )(𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )) 𝑗=1

𝑚+1

= 𝑓(𝜏𝑚+1 )𝛼(𝑡𝑚 ) − 𝑓(𝜏0 )𝛼(𝑡0 ) − ∑ 𝛼(𝑡𝑘−1 )(𝑓(𝜏𝑘 ) − 𝑓(𝜏𝑘−1 )) 𝑘=1

= 𝑓(𝑏)𝛼(𝑏) − 𝑓(𝑎)𝛼(𝑎) − 𝑆𝑓 (𝛼, 𝛱, 𝑃; [𝑎, 𝑏]) . Now, 𝜇(𝑃) → 0 implies 𝜇(𝛱) → 0, and Proposition 4.5 shows that in this case 𝑏

lim 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) = ∫ 𝑓(𝑥) 𝑑𝛼(𝑥)

𝜇(𝑃)→0

and

𝑎

𝑏

lim 𝑆𝑓 (𝛼, 𝛱, 𝑃; [𝑎, 𝑏]) = ∫ 𝛼(𝑥) 𝑑𝑓(𝑥)

𝜇(𝛱)→0

𝑎

which gives (4.55). Of course, in case 𝛼(𝑥) = 𝑥, equality (4.55) is nothing else but the classical integration by parts formula. In the proof of Proposition 4.24, we have used the hypotheses 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]) and 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) for technical reasons. If 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), but 𝑓 is only continuous, we may use formula (4.55) to define the RS-integral on the right-hand side. This gives yet another possibility of extending the RS-integral to larger classes of functions. The next proposition extends a well-known mean value theorem for the Riemann integral, see Exercise 4.23.

11 Here, we consider first 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } as a partition of [𝑎, 𝑏] and 𝛱 = {𝜏0 , 𝜏1 , . . . , 𝜏𝑚 , 𝜏𝑚+1 } as a set of intermediate points, and afterwards vice versa. This is possible since 𝜏𝑘−1 ≤ 𝑡𝑘−1 ≤ 𝜏𝑘 for 𝑘 = 1, 2, . . . , 𝑚 + 1.

4.1 Classical RS-integrals

|

291

Proposition 4.25 (mean value theorem). Let 𝑓 ∈ 𝐶([𝑎, 𝑏]), and let 𝛼 : [𝑎, 𝑏] → ℝ be monotonically increasing. Then there exists 𝜉 ∈ [𝑎, 𝑏] such that 𝑏

(4.56)

∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = 𝑓(𝜉)(𝛼(𝑏) − 𝛼(𝑎)) . 𝑎

Proof. Defining 𝑚(𝑓) as in (0.61) and 𝑀(𝑓) as in (0.62), from the estimate (4.13), we conclude that we may choose 𝜂 ∈ [𝑚(𝑓), 𝑀(𝑓)] with 𝑏

∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = 𝜂(𝛼(𝑏) − 𝛼(𝑎)) . 𝑎

Since 𝑓 is continuous, by the intermediate value theorem, we may find 𝜉 ∈ [𝑎, 𝑏] satisfying 𝑓(𝜉) = 𝜂 which proves the assertion. Proposition 4.25 is usually called the first mean value theorem for the RS-integral. An­ other mean value theorem may be found in Exercise 4.22. We close this section with an application of Proposition 4.24 to so-called gen­ eralized trapezoid inequalities for functions of bounded variation. Recall that the Dragomir trapezoid inequality [105] for functions 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) states that 󵄨󵄨󵄨 𝑏 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨∫ 𝛼(𝑡) 𝑑𝑡 − [𝛼(𝑏) − 𝛼(𝑎)] 𝑏 − 𝑎 󵄨󵄨󵄨 ≤ 1 (𝑏 − 𝑎) Var(𝛼; [𝑎, 𝑏]) . (4.57) 󵄨󵄨 󵄨󵄨 2 2 󵄨󵄨 𝑎 󵄨󵄨 󵄨 󵄨 More generally, in [77], the authors consider the function 𝑏

𝐽𝛼 (𝑥) := ∫ 𝛼(𝑡) 𝑑𝑡 − 𝛼(𝑎)(𝑥 − 𝑎) − 𝛼(𝑏)(𝑏 − 𝑥)

(𝑎 ≤ 𝑥 ≤ 𝑏) .

(4.58)

𝑎

Then the following estimate holds for the function 𝐽𝛼 . Proposition 4.26. For 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), the function (4.58) satisfies the estimate |𝐽𝛼 (𝑥)| ≤ max {𝑥 − 𝑎, 𝑏 − 𝑥} Var(𝛼; [𝑎, 𝑏]).

(4.59)

Proof. We claim that 𝑏

𝐽𝛼 (𝑥) = ∫(𝑥 − 𝑡) 𝑑𝛼(𝑡)

(𝑎 ≤ 𝑥 ≤ 𝑏) .

(4.60)

𝑎

Indeed, applying (4.55) for fixed 𝑥 ∈ [𝑎, 𝑏] to the function 𝑓(𝑡) := 𝑥 − 𝑡 and inte­ grating with respect to 𝑡, we obtain, by Theorem 4.17 , 𝑏

𝑏

∫(𝑥 − 𝑡) 𝑑𝛼(𝑡) = 𝑓(𝑏)𝛼(𝑏) − 𝑓(𝑎)𝛼(𝑎) − ∫ 𝛼(𝑡) 𝑑(𝑥 − 𝑡) 𝑎

𝑎 𝑏

= (𝑥 − 𝑏)𝛼(𝑏) − (𝑥 − 𝑎)𝛼(𝑎) + ∫ 𝛼(𝑡) 𝑑𝑡 = 𝐽𝛼 (𝑥) . 𝑎

292 | 4 Riemann–Stieltjes integrals Applying (4.45) to the RS-integral in (4.60) yields 󵄨󵄨 𝑏 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨∫(𝑥 − 𝑡) 𝑑𝛼(𝑡)󵄨󵄨󵄨 ≤ sup |𝑥 − 𝑡| Var(𝛼; [𝑎, 𝑏]) = max {𝑥 − 𝑎, 𝑏 − 𝑥} Var(𝛼; [𝑎, 𝑏]) 󵄨󵄨 󵄨󵄨 𝑎≤𝑡≤𝑏 󵄨󵄨 𝑎 󵄨󵄨 which proves the assertion. Putting, in particular, 𝑥 := (𝑏 + 𝑎)/2 in (4.59), we get precisely the trapezoidal formula (4.57). Moreover, applying (4.59) to the special choice 𝛼 := 𝜒(𝑎,𝑏) gives ‖𝛼‖𝐿 1 = 𝑏 − 𝑎 and Var(𝛼; [𝑎, 𝑏]) = 2, which shows that the constant 1/2 on the right-hand side of (4.57) is sharp.

4.2 Bounded variation and duality One of the numerous useful properties of the Riemann–Stieltjes integral is that it al­ lows us to rather easily describe the dual space of several well-known Banach spaces. Consider first the Banach space 𝑋 = 𝐶([𝑎, 𝑏]) of all continuous functions 𝑓 : [𝑎, 𝑏] → ℝ with the usual maximum norm (0.45). We want to associate with each ℓ ∈ 𝑋∗ a unique 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) in such a way that that 𝑋∗ becomes isomorphic to a subspace of 𝐵𝑉([𝑎, 𝑏]), and such that ℓ and 𝛼 are related by the formula 𝑏

⟨𝑓, ℓ⟩ = ⟨𝑓, ℓ𝛼 ⟩ = ∫ 𝑓(𝑡) 𝑑𝛼(𝑡)

(4.61)

𝑎

which we already encountered after the proof of Theorem 4.21. Here, we have adopted, as in Section 0.2, the notation ⟨𝑓, ℓ⟩ for ℓ(𝑓) to put evidence on the duality between 𝑋 and 𝑋∗ . First, we recall the concept of a normalized function of bounded variation, see Definition 1.2. A function 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) is normalized if 𝛼(𝑎) = 0 (i.e. 𝛼 belongs to the subspace 𝐵𝑉𝑜 ([𝑎, 𝑏])) and, in addition , 𝛼(𝑥0 ) = 𝛼(𝑥0 +) = lim 𝛼(𝑥) 𝑥→𝑥0 +

(4.62)

for 𝑎 ≤ 𝑥0 < 𝑏. In this case, we write 𝛼 ∈ 𝑁𝐵𝑉([𝑎, 𝑏]). For further use, we now discuss a procedure to associate to each 𝐵𝑉-function a normalized 𝐵𝑉-function. Definition 4.27. We associate to each 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) a function 𝛼# given by 0 { { { 𝛼#(𝑥) := {𝛼(𝑥+) − 𝛼(𝑎) { { {𝛼(𝑏) − 𝛼(𝑎) and call 𝛼# the (right) normalization of 𝛼.

for 𝑥 = 𝑎 , for 𝑎 < 𝑥 < 𝑏 ,

(4.63)

for 𝑥 = 𝑏 , ◼

4.2 Bounded variation and duality | 293

The name “normalization” is of course motivated by the fact that always 𝛼# ∈ 𝑁𝐵𝑉([𝑎, 𝑏]), and so 𝛼 󳨃→ 𝛼# defines a map from 𝐵𝑉([𝑎, 𝑏]) into 𝑁𝐵𝑉([𝑎, 𝑏]). Geo­ metrically, this map has two effects on 𝛼: it shifts the graph of 𝛼 vertically to fit the condition 𝛼#(𝑎) = 0, and it “fills the holes in the graph from the right” at any point where 𝛼 has a jump. In the next Proposition 4.28, we will summarize some interesting analytical properties of this map. To this end, we define an equivalence relation ≈ on 𝐵𝑉([𝑎, 𝑏]) by writing 𝛼 ≈ 𝛽 if ℓ𝛼 = ℓ𝛽 in (4.61), i.e. 𝑏

𝑏

∫ 𝑓(𝑡) 𝑑𝛼(𝑡) = ∫ 𝑓(𝑡) 𝑑𝛽(𝑡) 𝑎

𝑎

for all 𝑓 ∈ 𝐶([𝑎, 𝑏]). Proposition 4.28. The normalization (4.63) has the following properties. (a) We have 𝛼# ≈ 𝛼, i.e. every function 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) is equivalent to its normalization. (b) Each equivalence class with respect to ≈ contains exactly one normalized function. (c) The estimate Var(𝛼#; [𝑎, 𝑏]) ≤ Var(𝛼; [𝑎, 𝑏]) (4.64) holds. Proof. It is not hard to see that 𝛽 ≈ 0 if and only if 𝛽(𝑎) = 𝛽(𝑏) = 𝛽(𝑥+) = 𝛽(𝑥−)

(𝑎 < 𝑥 < 𝑏) .

(4.65)

Applying this to the function 𝛽 = 𝛼# − 𝛼, we get 𝛽(𝑎) = 𝛼#(𝑎) − 𝛼(𝑎) = −𝛼(𝑎) = 𝛼#(𝑏) − 𝛼(𝑏) = 𝛽(𝑏) and 𝛽(𝑥±) = 𝛼#(𝑥±) − 𝛼(𝑎) − 𝛼(𝑥±) = −𝛼(𝑎) for 𝑎 < 𝑥 < 𝑏, and so (a) is true. The statement (b) is of course an immediate conse­ quence of (a). To prove (c), let 𝜀 > 0, fix a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), and choose points 𝜏0 := 𝑎, 𝜏𝑚 := 𝑏, and 𝜏𝑗 ∈ (𝑡𝑗 , 𝑡𝑗+1 ) (𝑗 = 1, 2, . . . , 𝑚 − 1) at which 𝛼 is continuous and which are so close to 𝑡𝑗 that |𝛼(𝑡𝑗 +) − 𝛼(𝜏𝑗 )| < Then

𝑚

𝑚

𝑗=1

𝑗=1

𝜀 . 2𝑚

∑ |𝛼#(𝑡𝑗 ) − 𝛼#(𝑡𝑗−1 )| ≤ ∑ |𝛼(𝜏𝑗 ) − 𝛼(𝜏𝑗−1 )| + 𝜀 ≤ Var(𝛼) + 𝜀 ,

and so Var(𝛼#) ≤ Var(𝛼) + 𝜀. Since 𝜀 > 0 was arbitrary, we have proved that Var(𝛼#) ≤ Var(𝛼).

294 | 4 Riemann–Stieltjes integrals Proposition 4.28 shows, in particular, that the map 𝛼 󳨃→ 𝛼# is a bounded operator between 𝐵𝑉𝑜 ([𝑎, 𝑏]) and 𝑁𝐵𝑉([𝑎, 𝑏]) which satisfies ‖𝛼#‖𝐵𝑉 ≤ ‖𝛼‖𝐵𝑉 . As Exercise 4.47 shows, however, this map is not an isometry. The question arises regarding how to characterize the relation ≈ more explicitly, i.e. without referring to the integral (4.61). The following proposition gives a complete characterization which we formulate in terms of the set 𝛥(𝛼, 𝛽) = 𝛥(𝛼, 𝛽; [𝑎, 𝑏]) := {𝑥 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝛼(𝑥) ≠ 𝛽(𝑥)}

(4.66)

which we already introduced in (0.47): Proposition 4.29. For 𝛼, 𝛽 ∈ 𝐵𝑉([𝑎, 𝑏]), we have 𝛼 ≈ 𝛽 if and only if 𝛥(𝛼, 𝛽) is at most countable and contained in (𝑎, 𝑏). Proof. Suppose first that 𝛥(𝛼, 𝛽) = {𝑥1 , 𝑥2 , 𝑥3 , . . .} ⊂ (𝑎, 𝑏), and let 𝛾 := 𝛼 − 𝛽 and 𝑓 ∈ 𝐶([𝑎, 𝑏]). Then the real numbers 𝑎𝑛 := 𝛼(𝑥𝑛 ) − 𝛽(𝑥𝑛 ) (𝑛 = 1, 2, 3, . . .) satisfy

∞

∞

𝛾(𝑡) = ∑ 𝑎𝑛 (𝜒[𝑥𝑛 ,𝑏] (𝑡) − 𝜒(𝑥𝑛 ,𝑏] (𝑡)) , 𝑛=1

∑ |𝑎𝑛 | < ∞

𝑛=1

by construction. Applying Exercise 4.27 to this sequence (𝑎𝑛 )𝑛 (and 𝑏𝑛 := −𝑎𝑛 ), we con­ clude that 𝑏

∞

∫ 𝑓(𝑥) 𝑑𝛾(𝑥) = ∑ (𝑎𝑛 − 𝑎𝑛 )𝑓(𝑥𝑛 ) = 0 , 𝑛=1

𝑎

which implies 𝛾 ≈ 0, and thus 𝛼 ≈ 𝛽. Conversely, suppose now that 𝛾 ≈ 0 for some 𝛾 ∈ 𝐵𝑉([𝑎, 𝑏]) which means that 𝑏

∫ 𝑓(𝑥) 𝑑𝛾(𝑥) = 0 𝑎

for all 𝑓 ∈ 𝐶([𝑎, 𝑏]). We claim that 𝛾 is “almost constant” in the sense that the set {𝑥 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝛾(𝑥) ≠ 𝛾(𝑎)} is at most countable. Considering the function 𝑓(𝑥) ≡ 1, we see that 𝛾(𝑏) = 𝛾(𝑎). Since 𝛾 has bounded variation, the set 𝐷(𝛾) of all discontinuity points of 𝛾 in [𝑎, 𝑏] is at most countable. We show that 𝛾(𝑥) = 𝛾(𝑎) for all 𝑥 ∈ (𝑎, 𝑏)\𝐷(𝛾). In fact, given such a point 𝑥, choose 𝑛0 ∈ ℕ so large that 𝑛0 (𝑏 − 𝑥) > 1. For 𝑛 ≥ 𝑛0 , we define continuous functions 𝑓𝑛 : [𝑎, 𝑏] → ℝ by 1 for 𝑎 ≤ 𝑡 ≤ 𝑥 , { { { 𝑓𝑛 (𝑡) := {0 for 𝑥 + 𝑛1 ≤ 𝑡 ≤ 𝑏 , { { 1 {linear for 𝑥 ≤ 𝑡 ≤ 𝑥 + 𝑛 .

4.2 Bounded variation and duality | 295

By construction, then we have 𝑏

𝑥+1/𝑛

𝑥

0 = ∫ 𝑓𝑛 (𝑡) 𝑑𝛾(𝑡) = ∫ 𝑑𝛾(𝑡) + ∫ 𝑓𝑛 (𝑡) 𝑑𝛾(𝑡) 𝑎

𝑎

≤ 𝛾(𝑥) − 𝛾(𝑎) +

𝑥

max |𝑓𝑛 (𝑡)| Var(𝛾; [𝑥, 𝑥 + 1/𝑛])

𝑥≤𝑡≤𝑥+1/𝑛

= 𝛾(𝑥) − 𝛾(𝑎) + Var(𝛾; [𝑥, 𝑥 + 1/𝑛]) by Theorem 4.21 (a). However, Var(𝛾; [𝑥, 𝑥 + 1/𝑛]) → 0 (𝑛 → ∞), and so 𝛾(𝑥) = 𝛾(𝑎) for 𝑥 ∈ (𝑎, 𝑏) \ 𝐷(𝛾) as claimed. It is very easy to find an example which shows that the countability assumption on 𝛥(𝛼, 𝛽) in Proposition 4.29 cannot be dropped. The following example shows that the requirement 𝛥(𝛼, 𝛽) ⊆ (𝑎, 𝑏) cannot be dropped either. Example 4.30. On [𝑎, 𝑏] = [0, 1], let 𝑓(𝑥) ≡ 1, 𝛼(𝑥) ≡ 0, and 𝛽(𝑥) = 𝜒{1} (𝑥). Then 𝛥(𝛼, 𝛽) = {1} is finite, but 1

1

∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = 0 ≠ 1 = ∫ 𝑓(𝑥) 𝑑𝛽(𝑥), 0

0

♥

i.e. 𝛼 ≉ 𝛽.

Now, we are in a position to prove the announced duality theorem between the spaces 𝐶([𝑎, 𝑏]) and 𝑁𝐵𝑉([𝑎, 𝑏]). Theorem 4.31. The dual space 𝐶([𝑎, 𝑏])∗ of the space 𝐶([𝑎, 𝑏]) with the usual norm (4.67)

‖𝑓‖𝐶 = max |𝑓(𝑡)| 𝑎≤𝑡≤𝑏

may be identified with the space 𝑁𝐵𝑉([𝑎, 𝑏]). More precisely, the map 𝛷 : 𝑁𝐵𝑉([𝑎, 𝑏]) → 𝐶([𝑎, 𝑏])∗ ,

𝛷(𝛼) := ℓ𝛼 ,

with ℓ𝛼 as in (4.61) for 𝑓 ∈ 𝐶([𝑎, 𝑏]) and 𝛼 ∈ 𝑁𝐵𝑉([𝑎, 𝑏]), is a linear surjective isometry. Proof. The fundamental estimate (4.45) shows that 󵄨󵄨 𝑏 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 |⟨𝑓, ℓ𝛼 ⟩| = 󵄨󵄨∫ 𝑓(𝑡) 𝑑𝛼(𝑡)󵄨󵄨󵄨 ≤ ‖𝑓‖𝐶 ‖𝛼‖𝐵𝑉 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑎 󵄨󵄨 for 𝑓 ∈ 𝐶([𝑎, 𝑏]) and 𝛼 ∈ 𝐵𝑉𝑜 ([𝑎, 𝑏])), and so (4.61) indeed defines a bounded linear functional ℓ𝛼 on 𝑋 with ‖ℓ𝛼 ‖𝐶∗ ≤ ‖𝛼‖𝐵𝑉 . This means that 𝛷 is well-defined and satisfies ‖𝛷(𝛼)‖𝐶∗ ≤ ‖𝛼‖𝐵𝑉 .

296 | 4 Riemann–Stieltjes integrals The interesting and nontrivial part is of course to show that 𝛷 is surjective. So, let ℓ be an arbitrary bounded linear functional on 𝐶([𝑎, 𝑏]); by the Hahn–Banach theo­ rem (Theorem 0.21), we may extend ℓ to a bounded linear functional (which we also denote by ℓ) on the space 𝐵([𝑎, 𝑏]) with norm (0.39). We have to show that there exists a function 𝛼 ∈ 𝑁𝐵𝑉([𝑎, 𝑏])) such that ‖ℓ‖𝐶∗ = Var(𝛼) and ℓ = ℓ𝛼 . For 𝑎 < 𝑠 ≤ 𝑏, we put 𝑧𝑠 := 𝜒[𝑎,𝑠] , i.e. {1 if 𝑎 ≤ 𝑡 ≤ 𝑠 , 𝑧𝑠 (𝑡) := { 0 if 𝑠 < 𝑡 ≤ 𝑏 . { Obviously, 𝑧𝑠 ∈ 𝐵([𝑎, 𝑏]) with ‖𝑧𝑠 ‖∞ ≤ 1. In addition, for 𝑠 = 𝑎, we put 𝑧𝑎 (𝑡) ≡ 0. The function 𝛼 : [𝑎, 𝑏] → ℝ defined by 𝛼(𝑠) := ⟨𝑧𝑠 , ℓ⟩ satisfies both 𝛼(𝑎) = ⟨𝑧𝑎 , ℓ⟩ = 0 and (4.62); we claim that 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏])) and ‖𝛼‖𝐵𝑉 = Var(𝛼) ≤ ‖ℓ‖𝐶∗ .

(4.68)

To see this, fix a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]). We use the shortcut 𝜖𝑗 := 𝑠𝑔𝑛 [𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )]

(𝑗 = 1, 2, . . . , 𝑚)

and consider the step function {𝜖1 𝑤(𝑡) := { 𝜖 {𝑗 for 𝑗 = 2, 3, . . . , 𝑚. Since

for 𝑎 ≤ 𝑡 ≤ 𝑡1 , for 𝑡𝑗−1 < 𝑡 ≤ 𝑡𝑗

𝑚

𝑤(𝑡) = ∑ 𝜖𝑗 [𝑧𝑡𝑗 (𝑡) − 𝑧𝑡𝑗−1 (𝑡)] , 𝑗=1

applying the linear functional ℓ to 𝑤, we obtain 𝑚

⟨𝑤, ℓ⟩ = ∑ 𝜖𝑗 [⟨𝑧𝑡𝑗 , ℓ⟩ − ⟨𝑧𝑡𝑗−1 , ℓ⟩] 𝑗=1

𝑚 𝑚 󵄨 󵄨 = ∑ 𝜖𝑗 [𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )] = ∑ 󵄨󵄨󵄨󵄨𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )󵄨󵄨󵄨󵄨 , 𝑗=1

𝑗=1

by our definition of 𝜖𝑗 and 𝛼. However, |⟨𝑤, ℓ⟩| ≤ ‖ℓ‖𝐶∗ since ‖𝑤‖∞ ≤ 1, and so 𝑚 󵄨 󵄨 ∑ 󵄨󵄨󵄨󵄨𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )󵄨󵄨󵄨󵄨 ≤ ‖ℓ‖𝐶∗ .

(4.69)

𝑗=1

Since the right-hand side of (4.69) is independent of the partition 𝑃, we see that, in fact, 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏])) and Var(𝛼) ≤ ‖ℓ‖𝐶∗ as claimed. It remains to show that ℓ = ℓ𝛼 . With the same partition 𝑃 as before and 𝑓 ∈ 𝐶([𝑎, 𝑏]), we define 𝑧𝑃 : [𝑎, 𝑏] → ℝ by 𝑚

𝑧𝑃 (𝑡) := ∑ 𝑓(𝑡𝑗−1 ) [𝑧𝑡𝑗 (𝑡) − 𝑧𝑡𝑗−1 (𝑡)] . 𝑗=1

4.2 Bounded variation and duality | 297

A comparison with (4.17) shows that we may consider the value of ℓ at 𝑧𝑃 as a Riemann–Stieltjes sum 𝑚

⟨𝑧𝑃 , ℓ⟩ = ∑ 𝑓(𝑡𝑗−1 ) [𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )] = 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏])

(4.70)

𝑗=1

where¹² 𝛱 = 𝑃 \ {𝑡𝑚 } = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚−1 }. Moreover , {|𝑓(𝑎) − 𝑓(𝑡)| for 𝑎 ≤ 𝑡 ≤ 𝑡1 , |𝑧𝑃 (𝑡) − 𝑓(𝑡)| = { |𝑓(𝑡𝑗−1 ) − 𝑓(𝑡)| for 𝑡𝑗−1 < 𝑡 ≤ 𝑡𝑗 { for 𝑗 = 2, 3, . . . , 𝑚. Since 𝑓 is uniformly continuous on [𝑎, 𝑏], we have ‖𝑧𝑃 − 𝑓‖∞ → 0, and so ⟨𝑧𝑃 , ℓ⟩ → ⟨𝑓, ℓ⟩, as 𝜇(𝑃) → 0. However, we also know from Proposition 4.5 that 𝑏

⟨𝑓, ℓ⟩ = lim 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) = ∫ 𝑓(𝑡) 𝑑𝛼(𝑡) . 𝜇(𝑃)→0

(4.71)

𝑎

Thus, we have proved that ℓ = ℓ𝛼 = 𝛷(𝛼) which means that 𝛷 is surjective. Thus far, we have not made any assertion about the uniqueness of 𝛼, i.e. the injec­ tivity of 𝛷, and it is only here that we need the normalization (4.63) of 𝛼. In the first part of the proof, we have seen that ‖𝛷(𝛼)‖𝐶∗ = ‖ℓ𝛼 ‖𝐶∗ ≤ ‖𝛼‖𝐵𝑉 if ℓ𝛼 is given by (4.61). Since 𝛼 ≈ 𝛼#, by Proposition 4.28 (a), the integral in (4.61) does not change if we pass from the integrator 𝛼 to the integrator 𝛼#. So, we also have ‖𝛷(𝛼)‖𝐶∗ = ‖ℓ𝛼 ‖𝐶∗ ≤ ‖𝛼#‖𝐵𝑉 . On the other hand, in Proposition 4.28 (c), we have proved that ‖𝛼#‖𝐵𝑉 = Var(𝛼#) ≤ Var(𝛼) ≤ ‖ℓ𝛼 ‖𝐶∗ = ‖𝛷(𝛼)‖𝐶∗ . Thus, the map 𝛷 is an isometry, hence injective, and the proof is complete. It follows from the definition of the norm of a bounded linear functional (and our proof of Theorem 4.31 shows this as well) that 𝑏

} { (4.72) Var(𝛼; [𝑎, 𝑏]) = sup {∫ 𝑓(𝑥) 𝑑𝛼(𝑥) : 𝑓 ∈ 𝐶([𝑎, 𝑏]), ‖𝑓‖𝐶 ≤ 1} 𝑎 } { for all 𝛼 ∈ 𝑁𝐵𝑉([𝑎, 𝑏]). Conversely, the Hahn–Banach theorem (Corollary 0.22 (c)) im­ plies that 𝑏

{ } ‖𝑓‖𝐶 = sup {∫ 𝑓(𝑥) 𝑑𝛼(𝑥) : 𝛼 ∈ 𝑁𝐵𝑉([𝑎, 𝑏]), Var(𝛼; [𝑎, 𝑏]) ≤ 1} (4.73) 𝑎 { } for all 𝑓 ∈ 𝐶([𝑎, 𝑏]). This may also be proved quite easily by a direct computation:

12 More precisely, we take 𝛱 = {𝜏1 , 𝜏2 , . . . , 𝜏𝑚 } with 𝜏𝑗 := 𝑡𝑗−1 ; this set 𝛱 obviously satisfies (4.16).

298 | 4 Riemann–Stieltjes integrals Proposition 4.32. For any 𝑓 ∈ 𝐶([𝑎, 𝑏]), the equality (4.73) holds. Proof. Denoting the right-hand side of (4.73) by 𝑆, from (4.45), we immediately get the estimate ‖𝑓‖𝐶 ≥ 𝑆. To prove the reverse estimate, choose 𝜉 ∈ [𝑎, 𝑏] such that ‖𝑓‖𝐶 = |𝑓(𝜉)|. If 𝜉 < 𝑏, we define 𝛼 : [𝑎, 𝑏] → ℝ by if 𝑓(𝜉) > 0 and 𝜉 < 𝑥 ≤ 𝑏 , {1 { { 𝛼(𝑥) := 𝑠𝑔𝑛 𝑓(𝜉)𝜒(𝜉,𝑏] (𝑥) = {−1 if 𝑓(𝜉) < 0 and 𝜉 < 𝑥 ≤ 𝑏 , { { if 𝑓(𝜉) = 0 or 𝑎 ≤ 𝑥 ≤ 𝜉 . {0 Clearly, 𝛼 ∈ 𝑁𝐵𝑉([𝑎, 𝑏]) and Var(𝛼; [𝑎, 𝑏]) ≤ 1. Moreover , 𝑏

𝑆 ≥ ∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = |𝑓(𝜉)| = ‖𝑓‖𝐶 . 𝑎

In case 𝜉 = 𝑏, an analogous reasoning shows that 𝑆 ≥ |𝑓(𝑏 − 𝑡)| for any 𝑡 ∈ (0, 𝑏 − 𝑎], and the assertion follows from the continuity of 𝑓.

4.3 Bounded 𝑝-variation and duality Now, we are going to prove a parallel result to Theorem 4.31 by replacing 𝐵𝑉([𝑎, 𝑏]) by the spaces 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) for 1 < 𝑝 < ∞, see Definition 2.50. This result once more emphasizes the importance of these spaces. Recall that 𝑅𝐵𝑉𝑜𝑝 ([𝑎, 𝑏]) denotes the sub­ space of all 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) satisfying 𝑓(𝑎) = 0, see (3.70). Theorem 4.33. For 1 < 𝑝 < ∞, the dual space 𝐿 𝑝 ([𝑎, 𝑏])∗ of the space 𝐿 𝑝 ([𝑎, 𝑏]) with the usual norm 1/𝑝

𝑏 𝑝

‖𝑓‖𝐿 𝑝 = (∫ |𝑓(𝑡)| 𝑑𝑡)

(4.74)

𝑎

may be identified with the space 𝑅𝐵𝑉𝑜𝑝󸀠 ([𝑎, 𝑏]), where 𝑝󸀠 = 𝑝/(𝑝 − 1). More precisely, the map 𝛷 : 𝑅𝐵𝑉𝑜𝑝󸀠 ([𝑎, 𝑏]) → 𝐿 𝑝 ([𝑎, 𝑏])∗ , 𝛷(𝛼) := ℓ𝛼 , with ℓ𝛼 as in (4.61) for 𝑓 ∈ 𝐿 𝑝 ([𝑎, 𝑏]) and 𝛼 ∈ 𝑅𝐵𝑉𝑜𝑝󸀠 ([𝑎, 𝑏]), is a linear surjective isometry. Proof. The proof is quite similar to that of the preceding Theorem 4.31, so we point out only the differences. One technical advantage is that we need not normalize 𝛼 now since functions from 𝑅𝐵𝑉𝑝󸀠 ([𝑎, 𝑏]) are continuous for 𝑝󸀠 > 1. Given 𝑓 ∈ 𝐿 𝑝 ([𝑎, 𝑏]), 𝛼 ∈ 𝑅𝐵𝑉𝑜𝑝󸀠 ([𝑎, 𝑏]), and a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), we define 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) as in (4.70). By Hölder’s inequality (0.108), we then

4.3 Bounded 𝑝-variation and duality

| 299

get 𝑚

|𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏])| = ∑ |𝑓(𝑡𝑗−1 | |𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )| 𝑗=1

𝑚

≤ (∑ 𝑗=1

|𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )|𝑝 󸀠

|𝑡𝑗 − 𝑡𝑗−1 |𝑝 −1 󸀠

1/𝑝󸀠

󸀠

)

1/𝑝

𝑚

𝑝

(∑ |𝑓(𝑡𝑗−1 )| |𝑡𝑗 − 𝑡𝑗−1 |)

(4.75)

𝑗=1

1/𝑝

𝑚

≤ Var𝑅𝑝󸀠 (𝛼)1/𝑝 (∑ |𝑓(𝑡𝑗−1 )|𝑝 |𝑡𝑗 − 𝑡𝑗−1 |)

.

𝑗=1

Now, if 𝑓 is continuous on [𝑎, 𝑏], then the limit (4.71) exists, and (4.75) together with 𝛼(𝑎) = 0 shows that 1/𝑝 󵄨󵄨 󵄨󵄨 𝑏 𝑏 󵄨󵄨 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨 𝑅 1/𝑝 (∫ |𝑓(𝑡)|𝑝 𝑑𝑡) = ‖𝑓‖𝐿 𝑝 ‖𝛼‖𝑅𝐵𝑉 󸀠 . 󵄨󵄨󵄨∫ 𝑓(𝑡) 𝑑𝛼(𝑡)󵄨󵄨󵄨 ≤ Var𝑝󸀠 (𝛼) 𝑝 󵄨󵄨 󵄨󵄨 𝑎 𝑎 󵄨 󵄨

(4.76)

Since 𝐶([𝑎, 𝑏]) is dense in 𝐿 𝑝 ([𝑎, 𝑏]) with respect to the norm (4.74), there exists a unique extension of the integral (4.61) to the entire space 𝐿 𝑝 ([𝑎, 𝑏]), and we use the same notation for this extension. Thus, we have proved that the map 𝛷 is well-defined and satisfies ‖𝛷(𝛼)‖𝐿∗𝑝 ≤ ‖𝛼‖𝑅𝐵𝑉𝑝󸀠 . To show that 𝛷 is surjective, we proceed as in the proof of Theorem 4.31. Thus, given an arbitrary bounded linear functional ℓ on 𝐿 𝑝 ([𝑎, 𝑏]), we have to find a 𝛼 ∈ 𝑅𝐵𝑉𝑝󸀠 ([𝑎, 𝑏])) such that 󸀠

‖ℓ‖𝐿∗𝑝 = Var𝑅𝑝󸀠 (𝛼)1/𝑝 ,

𝛼(𝑎) = 0,

ℓ = ℓ𝛼 .

Defining 𝑧𝑠 as before, we have 𝑧𝑠 ∈ 𝐿 𝑝 ([𝑎, 𝑏]) with ‖𝑧𝑠 ‖𝐿 𝑝 = 𝑠1/𝑝 . Putting again 𝛼(𝑠) := ⟨𝑧𝑠 , ℓ⟩ and fixing a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), we obtain |𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )| = |⟨𝑧𝑡𝑗 , ℓ⟩ − ⟨𝑧𝑡𝑗−1 , ℓ⟩| ≤ ‖ℓ‖𝐿∗𝑝 ‖𝑧𝑡𝑗 − 𝑧𝑡𝑗−1 ‖𝐿 𝑝 1/𝑝

𝑏

󵄨󵄨𝑝 󵄨󵄨 = ‖ℓ‖𝐿∗𝑝 (∫ 󵄨󵄨󵄨𝜒[𝑎,𝑡𝑗 ] (𝑡) − 𝜒[𝑎,𝑡𝑗−1 ] (𝑡)󵄨󵄨󵄨 𝑑𝑡) 󵄨 󵄨 𝑎

= ‖ℓ‖𝐿∗𝑝 |𝑡𝑗 − 𝑡𝑗−1 | ,

and hence 𝑚

∑ 𝑗=1

since

|𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )|𝑝 |𝑡𝑗 − 𝑡𝑗−1

󸀠 |𝑝 −1

󸀠

𝑝󸀠

𝑚

󸀠

𝑝󸀠

≤ ‖ℓ‖𝐿∗ ∑ |𝑡𝑗 − 𝑡𝑗−1 |𝑝 /𝑝−1/(𝑝−1) = ‖ℓ‖𝐿∗ 𝑝

𝑗=1

𝑝 𝑝󸀠 1 1 − = − = 0. 𝑝 𝑝 − 1 𝑝(𝑝 − 1) 𝑝 − 1

𝑝

300 | 4 Riemann–Stieltjes integrals This shows that 𝛼 ∈ 𝑅𝐵𝑉𝑝󸀠 ([𝑎, 𝑏])) and ‖𝛼‖𝑅𝐵𝑉𝑝󸀠 ≤ ‖ℓ‖𝐿∗𝑝 . Now, we prove that ‖𝛼‖𝑅𝐵𝑉𝑝󸀠 = ‖ℓ‖𝐿∗𝑝 . In fact, all functions of the form 𝑚

𝑓(𝑡) := ∑ 𝜉𝑗 𝜒[𝑡𝑗−1 ,𝑡𝑗 )

(4.77)

(𝜉1 , 𝜉2 , . . . , 𝜉𝑚 ∈ ℝ)

𝑗=1

belong to 𝐿 𝑝 ([𝑎, 𝑏]) and satisfy 1/𝑝

𝑚

𝑝

‖𝑓‖𝐿 𝑝 = (∑ |𝜉𝑗 | |𝑡𝑗 − 𝑡𝑗−1 |)

.

𝑗=1

Applying the functional ℓ to such a function and using (0.108) yields 𝑚

|⟨𝑓, ℓ⟩| ≤ ∑ |𝜉𝑗 | |𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )| 𝑗=1

𝑚

≤ (∑

|𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )|𝑝 󸀠

|𝑡𝑗 − 𝑡𝑗−1 |𝑝 −1

𝑗=1

󸀠

1/𝑝󸀠

)

𝑚

1/𝑝 𝑝

( ∑ |𝜉𝑗 | |𝑡𝑗 − 𝑡𝑗−1 |

(𝑝󸀠 −1)𝑝/𝑝󸀠

)

,

𝑗=1

󸀠

so |⟨𝑓, ℓ⟩| ≤ Var𝑅𝑝󸀠 (𝛼)1/𝑝 ‖𝑓‖𝐿 𝑝 . Since the set of functions of the form (4.77) is dense in 󸀠

𝐿 𝑝 ([𝑎, 𝑏]), we conclude that ‖ℓ‖𝐿∗𝑝 ≤ Var𝑅𝑝󸀠 (𝛼)1/𝑝 = ‖𝛼‖𝑅𝐵𝑉𝑝󸀠 . It remains to prove (4.61). Choosing 𝑓 = 𝑧𝑠 in (4.71) gives 𝑏

lim 𝑆𝛼 (𝑧𝑠 , 𝑃, 𝛱; [𝑎, 𝑏]) = ∫ 𝑧𝑠 (𝑡) 𝑑𝛼(𝑡) = ⟨𝑧𝑠 , ℓ𝛼 ⟩ ,

𝜇(𝑃)→0

𝑎

where we may consider, without loss of generality, only partitions 𝑃 which contain the point 𝑠. For such partitions, we have, by (4.70) , 𝑚

𝑆𝛼 (𝑧𝑠 , 𝑃, 𝛱; [𝑎, 𝑏]) = ∑ [𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )] 𝜒[𝑎,𝑠] (𝑡𝑗−1 ) = 𝛼(𝑠) − 𝛼(𝑎) = 𝛼(𝑠) , 𝑗=1

and hence, after passing to the limit as 𝜇(𝑃) → 0 , 𝑏

∫ 𝑧𝑠 (𝑡) 𝑑𝛼(𝑡) = 𝛼(𝑠) . 𝑎

Applying this to functions 𝑓 of the form (4.77), we get 𝑏

𝑚

⟨𝑓, ℓ⟩ = ∫ 𝑓(𝑡) 𝑑𝛼(𝑡) = ∑ [𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )] 𝜉𝑗 . 𝑎

𝑗=1

Again, since the set of functions of the form (4.77) is dense in 𝐿 𝑝 ([𝑎, 𝑏]), we have proved (4.61) for all 𝑓 ∈ 𝐿 𝑝 ([𝑎, 𝑏]). Thus, we may put 𝛷(𝛼) = ℓ, and the proof of the surjectivity of 𝛷 is complete. The injectivity of 𝛷 follows from the isometry property ‖𝛷(𝛼)‖𝐿∗𝑝 = ‖𝛼‖𝑅𝐵𝑉𝑝󸀠 .

4.3 Bounded 𝑝-variation and duality

| 301

Theorem 4.33 refers to the case 𝑝 > 1. One could expect that a parallel result holds in case 𝑝 = 1, where 𝑅𝐵𝑉𝑜∞ ([𝑎, 𝑏])) has to be interpreted as space 𝐿𝑖𝑝𝑜 ([𝑎, 𝑏]) of all Lipschitz continuous functions 𝛼 : [𝑎, 𝑏] → ℝ with 𝛼(𝑎) = 0. In fact, for 𝑓 ∈ 𝐿 1 ([𝑎, 𝑏]), we obtain the estimate 󵄨󵄨 󵄨󵄨 𝑏 𝑏 󵄨󵄨 󵄨󵄨 |𝛼(𝑥) − 𝛼(𝑦)| 󵄨 󵄨󵄨 ∫ |𝑓(𝑡)| 𝑑𝑡 = ‖𝑓‖𝐿 1 ‖𝛼‖𝐿𝑖𝑝 , 󵄨󵄨∫ 𝑓(𝑡) 𝑑𝛼(𝑡)󵄨󵄨󵄨 ≤ sup 󵄨󵄨 𝑥=𝑦̸ 󵄨󵄨 |𝑥 − 𝑦| 󵄨󵄨 󵄨󵄨 𝑎 𝑎

(4.78)

which is the precise analogue to (4.76). The remaining part of the proof carries over; we summarize with the following Theorem 4.34. The dual space 𝐿 1 ([𝑎, 𝑏])∗ of the space 𝐿 1 ([𝑎, 𝑏]) with norm 𝑏

(4.79)

‖𝑓‖𝐿 1 = ∫ |𝑓(𝑡)| 𝑑𝑡 𝑎

may be identified with the space 𝑅𝐵𝑉𝑜∞ ([𝑎, 𝑏]) = 𝐿𝑖𝑝𝑜 ([𝑎, 𝑏]) of Lipschitz continuous functions 𝛼 : [𝑎, 𝑏] → ℝ satisfying 𝛼(𝑎) = 0. More precisely, the map 𝛷 : 𝐿𝑖𝑝𝑜 ([𝑎, 𝑏]) → 𝐿 1 ([𝑎, 𝑏])∗ ,

𝛷(𝛼) := ℓ𝛼 ,

with ℓ𝛼 as in (4.61) for 𝑓 ∈ 𝐿 1 ([𝑎, 𝑏]) and 𝛼 ∈ 𝐿𝑖𝑝𝑜 ([𝑎, 𝑏])), is a linear surjective isometry. The Theorems 4.33 and 4.34 are somewhat unsatisfactory, insofar as they connect the Riesz space 𝑅𝐵𝑉𝑝󸀠 to the Lebesgue space 𝐿 𝑝 which is quite different in nature. How­ ever, the Riesz theorem (Theorem 3.34) allows us to establish a more “intrinsic” duality inside the scale of 𝑅𝐵𝑉𝑝 -spaces. Indeed, from equality (3.69), it follows that for 1 < 𝑝 < ∞, the differential operator 𝐷𝑓 := 𝑓󸀠 defines a linear surjective isometry between the spaces (𝑅𝐵𝑉𝑜𝑝 , ‖ ⋅ ‖𝑅𝐵𝑉𝑝 ) and (𝐿 𝑝 , ‖⋅‖𝐿 𝑝 ). So, we may combine this with the duality result established in Theorem 4.33 to get the following more satisfactory version: Theorem 4.35. For 1 < 𝑝 < ∞, the dual space 𝑅𝐵𝑉𝑜𝑝 ([𝑎, 𝑏])∗ of the space 𝑅𝐵𝑉𝑜𝑝 ([𝑎, 𝑏]) with norm (2.90) may be identified with the space 𝑅𝐵𝑉𝑜𝑝󸀠 ([𝑎, 𝑏]), where 𝑝󸀠 = 𝑝/(𝑝 − 1). More precisely, the map 𝛹 : 𝑅𝐵𝑉𝑜𝑝󸀠 ([𝑎, 𝑏]) → 𝑅𝐵𝑉𝑜𝑝 ([𝑎, 𝑏])∗ , with

𝛹(𝛼) := ℓ𝛼 ,

𝑏

⟨𝑓, ℓ𝛼 ⟩ := ∫ 𝑓󸀠 (𝑡) 𝑑𝛼(𝑡) 𝑎

for 𝑓 ∈

𝑅𝐵𝑉𝑜𝑝 ([𝑎, 𝑏])

and 𝛼 ∈

𝑅𝐵𝑉𝑜𝑝󸀠 ([𝑎, 𝑏]),

is a linear surjective isometry.

302 | 4 Riemann–Stieltjes integrals

4.4 Nonclassical RS-integrals The classical Riemann–Stieltjes integral has been generalized in various directions. Some of these generalizations aim at avoiding the nonexistence of the integral (4.7) due to the presence of common points of discontinuity of 𝑓 and 𝛼, see Theorem 4.15. We will mention some results of this type in the next section. Other extensions of the RS-integral (4.7) consist of replacing the class 𝐵𝑉([𝑎, 𝑏]) with larger classes related to higher order variations Var𝑊 𝑘 (𝛼; [𝑎, 𝑏]) (Section 2.7), or the Waterman class Λ𝐵𝑉([𝑎, 𝑏]) discussed in Section 2.2, or the Schramm class 𝛷𝐵𝑉([𝑎, 𝑏]) discussed in Section 2.3. We will describe the corresponding procedure now for the space 𝛷𝐵𝑉([𝑎, 𝑏]); unfortunately, this requires a great deal of technical definitions and results. Our main theorem in this section asserts that under suitable growth conditions on two Schramm sequences 𝛷 = (𝜙𝑛 )𝑛 and 𝛹 = (𝜓𝑛 )𝑛 , the integral (4.7) exists for 𝑓 ∈ 𝛷𝐵𝑉([𝑎, 𝑏])∩𝐶([𝑎, 𝑏]) and 𝛼 ∈ 𝛹𝐵𝑉([𝑎, 𝑏]), for a precise formulation, see Theorem 4.40 below. The whole material of this section is taken from Schramm’s survey paper [286]. Recall that 𝛷 = (𝜙𝑛 )𝑛 is called a Schramm sequence if each 𝜙𝑛 : [0, ∞) → [0, ∞) is a convex Young function and 𝜙𝑛+1 (𝑥) ≤ 𝜙𝑛 (𝑥) for all 𝑛 ∈ ℕ. If, in addition, ∞

∑ 𝜙𝑛 (𝑥) = ∞ (𝑥 > 0) ,

(4.80)

𝑛=1

then 𝛷 = (𝜙𝑛 )𝑛 is called a divergent Schramm sequence. The space 𝛷𝐵𝑉([𝑎, 𝑏]) con­ sists, by definition, of all functions 𝑓 : [𝑎, 𝑏] → ℝ satisfying Var𝛷 (𝑐𝑓) < ∞ for some 𝑐 > 0, where Var𝛷 (𝑓) = Var𝛷 (𝑓; [𝑎, 𝑏]) ∞

= sup { ∑ 𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) : {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏])}

(4.81)

𝑛=1

and the supremum in (4.81) is taken over all infinite collections in 𝛴∞ ([𝑎, 𝑏]), see Sec­ tion 1.2. This space, equipped with the norm ‖𝑓‖𝛷 = |𝑓(𝑎)| + inf {𝜆 > 0 : Var𝛷 (𝑓/𝜆) ≤ 1} ,

(4.82)

is a Banach space; in particular, the infimum in (4.82) is a norm on the subspace 𝛷𝐵𝑉𝑜 ([𝑎, 𝑏]) of all 𝑓 ∈ 𝛷𝐵𝑉([𝑎, 𝑏]) satisfying 𝑓(𝑎) = 0. To introduce a Riemann–Stieltjes integral (4.7) for 𝛼 belonging to 𝛷𝐵𝑉([𝑎, 𝑏]), we need a series of technical lemmas. We begin with the following Lemma 4.36. Let 𝛷 = (𝜙𝑛 )𝑛 and 𝛹 = (𝜓𝑛 )𝑛 be two Schramm sequences satisfying ∞

∑ 𝜙𝑘−1 (1/𝑘)𝜓𝑘−1 (1/𝑘) < ∞ .

𝑘=1

Then the following holds.

(4.83)

4.4 Nonclassical RS-integrals

|

303

(a) If 𝐴 and 𝐵 are any nonnegative real numbers, then ∞

∑ 𝜙𝑘−1 (𝐴/𝑘)𝜓𝑘−1 (𝐵/𝑘) < ∞ .

(4.84)

𝑘=1

(b) There exists a convex Young function 𝛤 : [0, ∞) → [0, ∞) satisfying 𝛤(𝑥) = 𝑜(𝑥) as 𝑥 → 0 and ∞

∑ (𝛤 ∘ 𝜙𝑘 )−1 (1/𝑘)(𝛤 ∘ 𝜓𝑘 )−1 (1/𝑘) < ∞ .

(4.85)

𝑘=1

Proof. (a) Choose 𝑚 ∈ ℕ such that 𝑚 ≥ 𝐴 and 𝑚 ≥ 𝐵. For 𝑗𝑚 ≤ 𝑘 < (𝑗 + 1)𝑚, we then have ∞

∑ 𝜙𝑘−1 (𝐴/𝑘)𝜓𝑘−1 (𝐵/𝑘)

𝑘=1

𝑚−1

∞

𝑘=1

𝑗=1

(4.86)

≤ ∑ 𝜙𝑘−1 (𝐴/𝑘)𝜓𝑘−1 (𝐵/𝑘) + 𝑚 ∑ 𝜙𝑗−1 (1/𝑗)𝜓𝑗−1 (1/𝑗) < ∞ since both 𝜙𝑘−1 and 𝜓𝑘−1 are increasing. (b) From (a) it follows that ∞

∑ 𝜙𝑘−1 (3𝑛/𝑘)𝜓𝑘−1 (3𝑛/𝑘) < ∞

(𝑛 = 1, 2, 3, . . .) .

𝑘=1

Thus, we may choose a positive increasing sequence (𝑘𝑛 )𝑛 of natural numbers so that 1 𝑘𝑛+1 > (1 + ) 𝑘𝑛 , 𝑛

∞

∑ 𝜙𝑘−1 (3𝑛/𝑘)𝜓𝑘−1 (3𝑛/𝑘)


0 ,

(4.87)

for 𝑥 = 0 .

Then 𝜒𝑛 (𝛤, 𝛷; 𝑥) ≤ 𝜒𝑛+1 (𝛤, 𝛷; 𝑥) for 𝑥 ≥ 0 and 𝜒𝑛 (𝛤, 𝛷; 𝑥) → 0 as 𝑥 → 0. Moreover, denoting by osc(𝑓; [𝑎, 𝑏]) the oscillation (1.12) of 𝑓 on [𝑎, 𝑏], from the trivial estimate

4.4 Nonclassical RS-integrals

|

305

𝜙1 (osc(𝑓; [𝑎, 𝑏])) ≤ Var𝛷 (𝑓), we get ∞

Var𝛤∘𝛷 (𝑓) = sup { ∑ 𝛤(𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|))} 𝑛=1 ∞

= sup { ∑ 𝜒𝑛 (𝛤, 𝛷; |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|)𝜙𝑛 (|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|)}

(4.88)

𝑛=1

≤ 𝜒1 (𝛤, 𝛷; osc(𝑓 : [𝑎, 𝑏])) Var𝛷 (𝑓) where both suprema are taken over all collections {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]). This again shows that 𝛷𝐵𝑉([𝑎, 𝑏]) ⊆ 𝛤𝛷𝐵𝑉([𝑎, 𝑏]). We now introduce some notation which will simplify our computations. Given a Schramm sequence 𝛷 = (𝜙𝑛 )𝑛 and a sequence 𝐴 = (𝑎𝑛 )𝑛 of real numbers, we use the shortcut 𝑛

𝜌𝑛 (𝛷; 𝐴) := ∑ 𝜙𝑘 (|𝑎𝑘 |)

(𝑛 = 1, 2, 3, . . .) .

(4.89)

𝑘=1

Moreover, we denote by I the family of all strictly increasing sequences 𝐼 = (𝑖𝑛 )𝑛 starting from 𝑖0 ≥ 0. We associate to each pair of sequences 𝐴 = (𝑎𝑛 )𝑛 and 𝐼 = (𝑖𝑛 )𝑛 ∈ I another sequence 𝐼(𝐴) with terms 𝑖1

𝑖2

𝑖𝑛

𝐼(𝐴) := ( ∑ 𝑎𝑘 , ∑ 𝑎𝑘 , . . . , ∑ 𝑘=𝑖0 +1

𝑘=𝑖1 +1

𝑎𝑘 , . . .) .

(4.90)

𝑘=𝑖𝑛−1 +1

Loosely speaking, the sequence 𝐼(𝐴) is derived from the sequence 𝐴 by “adding its terms sectionwise” according to the sections prescribe by the index set 𝐼. For example, if 𝐼 is the set of prime numbers, then 𝐼(𝐴) = (𝑎1 + 𝑎2 , 𝑎3 , 𝑎4 + 𝑎5 , 𝑎6 + 𝑎7 , 𝑎8 + 𝑎9 + 𝑎10 + 𝑎11 , 𝑎12 + 𝑎13 , . . .) . We also put 𝜌𝑛∗ (𝛷; 𝐴) := sup {𝜌𝑛 (𝛷; 𝐼(𝐴)) : 𝐼 ∈ I} ,

(4.91)

where the supremum in (4.91) is taken over all sequences 𝐼 ∈ I. Using this notation, we now have the following Lemma 4.37. Let 𝐴 = (𝑎𝑛 )𝑛 and 𝐵 = (𝑏𝑛 )𝑛 be real sequences, and let 𝛷 = (𝜙𝑛 )𝑛 and 𝛹 = (𝜓𝑛 )𝑛 be two Schramm sequences. Then the following is true. (a) The estimate 𝜌 (𝛷; 𝐼(𝐴)) √𝑛 |𝑎1 𝑎2 ⋅ ⋅ ⋅ 𝑎𝑛 | ≤ 𝜙𝑛−1 ( 𝑛 ) (4.92) 𝑛 holds for all 𝑛 ∈ ℕ. (b) For every 𝑛 ∈ ℕ, there exists a 𝑘0 ∈ ℕ, 1 ≤ 𝑘0 ≤ 𝑛, such that |𝑎𝑘0 𝑏𝑘0 | ≤ 𝜙𝑛−1 (

𝜌𝑛 (𝛷; 𝐼(𝐴)) 𝜌 (𝛹; 𝐼(𝐵)) ) 𝜓𝑛−1 ( 𝑛 ). 𝑛 𝑛

(4.93)

306 | 4 Riemann–Stieltjes integrals Proof. By the arithmetic-geometric mean inequality and the monotonicity of 𝜙𝑛 , we have 1 𝜙𝑛 (√𝑛 |𝑎1 𝑎2 ⋅ ⋅ ⋅ 𝑎𝑛 |) ≤ 𝜙𝑛 ( (|𝑎1 | + |𝑎2 | + . . . + |𝑎𝑛 |)) . 𝑛 Moreover, from Jensen’s inequality (2.18), it follows that 1 𝑛 1 𝑛 1 𝜙𝑛 ( ∑ |𝑎𝑘 |) ≤ ∑ 𝜙𝑘 (|𝑎𝑘 |) = 𝜌𝑛 (𝛷; 𝐼(𝐴)) 𝑛 𝑘=1 𝑛 𝑘=1 𝑛 and applying 𝜙𝑛−1 , we obtain (a). To prove (b), we choose 𝑘0 ∈ {1, 2, . . . , 𝑛} such that |𝑎𝑘0 𝑏𝑘0 | = min {|𝑎𝑘 𝑏𝑘 | : 𝑘 = 1, 2, . . . , 𝑛}. Then 󵄨󵄨 𝑛 󵄨󵄨1/𝑛 󵄨󵄨 𝑛 󵄨󵄨1/𝑛 󵄨󵄨 𝑛 󵄨󵄨1/𝑛 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 |𝑎𝑘0 𝑏𝑘0 | ≤ 󵄨󵄨∏ 𝑎𝑘 𝑏𝑘 󵄨󵄨󵄨 = 󵄨󵄨󵄨∏ 𝑎𝑘 󵄨󵄨󵄨 󵄨󵄨󵄨∏ 𝑏𝑘 󵄨󵄨󵄨 󵄨󵄨 𝑘=1 󵄨󵄨 𝑘=1 󵄨󵄨 󵄨󵄨 𝑘=1 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (𝛷; 𝐼(𝐴)) (𝛹; 𝐼(𝐵)) 𝜌 𝜌 ) 𝜓𝑛−1 ( 𝑛 ), ≤ 𝜙𝑛−1 ( 𝑛 𝑛 𝑛

(4.94)

where in the last inequality we have used (a). Before we state the main result of this section, we need another two auxiliary propo­ sitions. We start with the following refinement of Lemma 4.37 (b). Proposition 4.38. Under the hypotheses of Lemma 4.37, the estimate 󵄨󵄨 󵄨󵄨 𝑛 𝑘 󵄨 󵄨󵄨 󵄨󵄨 ∑ ∑ 𝑎𝑗 𝑏𝑘 󵄨󵄨󵄨 ≤ 𝜙−1 (𝜌∗ (𝛷; 𝐼(𝐴))) 𝜓−1 (𝜌∗ (𝛹; 𝐼(𝐵))) 1 1 1 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑘=1 𝑖=1 𝑛−1 𝜌∗ (𝛷; 𝐼(𝐴)) 𝜌∗ (𝛹; 𝐼(𝐵)) ) 𝜓𝑘−1 ( 𝑘 ) + ∑ 𝜙𝑘−1 ( 𝑘 𝑘 𝑘 𝑘=1

(4.95)

is true.¹³ Proof. We prove (4.95) by induction over 𝑛. For 𝑛 = 1, we have |𝑎1 𝑏1 | = 𝜙1−1 (𝜙1 (|𝑎1 |))𝜓1−1 (𝜓1 (|𝑏1 |)) ≤ 𝜙1−1 (𝜌1∗ (𝛷; 𝐼(𝐴))) 𝜓1−1 (𝜌1∗ (𝛹; 𝐼(𝐵))) . Suppose that 𝑛 ≥ 2 and (4.95) holds true for 𝑛 − 1. Defining a shifted sequence 𝐴󸀠 = (𝑎𝑛󸀠 )𝑛 by 𝑎𝑘󸀠 := 𝑎𝑘+1 , by Lemma 4.37 (b), we find a 𝑘0 ∈ {1, 2, . . . , 𝑛 − 1} such that −1 ( |𝑎𝑘0 +1 𝑏𝑘0 | ≤ 𝜙𝑛−1

𝜌𝑛−1 (𝛷; 𝐼(𝐴󸀠 )) 𝜌 (𝛹; 𝐼(𝐵)) −1 ) 𝜓𝑛−1 ). ( 𝑛−1 𝑛−1 𝑛−1

From ∗ (𝛷; 𝐼(𝐴󸀠 )) ≤ 𝜌𝑛−1 (𝛷; 𝐼(𝐴)) 𝜌𝑛−1 (𝛷; 𝐼(𝐴󸀠 )) ≤ 𝜌𝑛−1

13 In case 𝑛 = 1, we take the last sum in (4.95) to be 0.

4.4 Nonclassical RS-integrals

| 307

and ∗ (𝛹; 𝐼(𝐵)), 𝜌𝑛−1 (𝛹; 𝐼(𝐵)) ≤ 𝜌𝑛−1

it follows that −1 |𝑎𝑘0 +1 𝑏𝑘0 | ≤ 𝜙𝑛−1 (

∗ 𝜌𝑛−1 𝜌∗ (𝛹; 𝐼(𝐵)) (𝛷; 𝐼(𝐴)) −1 ) 𝜓𝑛−1 ). ( 𝑛−1 𝑛−1 𝑛−1

(4.96)

To reduce our claim to the induction hypothesis, we define two sequences 𝐶 = (𝑐𝑛 )𝑛 and 𝐷 = (𝑑𝑛 )𝑛 by {𝑎𝑘 { { 𝑐𝑘 := {𝑎𝑘0 + 𝑎𝑘0 +1 { { {𝑎𝑘+1

for 𝑘 < 𝑘0 ,

𝑏𝑘 { { { 𝑑𝑘 := {𝑏𝑘0 + 𝑏𝑘0 +1 { { {𝑏𝑘+1

for 𝑘 < 𝑘0 ,

for 𝑘 = 𝑘0 , for 𝑘 > 𝑘0 ,

and for 𝑘 = 𝑘0 , for 𝑘 > 𝑘0 ,

respectively, and note that 𝜌𝑛∗ (𝛷; 𝐼(𝐶)) ≤ 𝜌𝑛∗ (𝛷; 𝐼(𝐴)) and 𝜌𝑛∗ (𝛹; 𝐼(𝐷)) ≤ 𝜌𝑛∗ (𝛹; 𝐼(𝐵)). A straightforward calculation shows that 𝑛−1 𝑘

𝑛

𝑘

∑ ∑ 𝑐𝑗 𝑑𝑘 = 𝑎𝑘0 +1 𝑏𝑘0 + ∑ ∑ 𝑎𝑗 𝑏𝑘 .

𝑘=1 𝑖=1

𝑘=1 𝑖=1

Thus, combining this with (4.96), we obtain 󵄨󵄨𝑛−1 𝑘 󵄨󵄨 𝑛 𝑘 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 ∑ ∑ 𝑎𝑗 𝑏𝑘 󵄨󵄨󵄨 ≤ |𝑎𝑘 +1 𝑏𝑘 | + 󵄨󵄨󵄨 ∑ ∑ 𝑐𝑗 𝑑𝑘 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 0 󵄨󵄨𝑘=1 𝑖=1 󵄨󵄨𝑘=1 𝑖=1 󵄨󵄨 󵄨󵄨 −1 ∗ −1 ∗ ≤ 𝜙1 (𝜌1 (𝛷; 𝐼(𝐶)))𝜓1 (𝜌1 (𝛹; 𝐼(𝐷))) 𝑛−2

+ ∑ 𝜙𝑘−1 ( 𝑘=1

−1 + 𝜙𝑛−1 (

𝜌𝑘∗ (𝛷; 𝐼(𝐶)) 𝜌∗ (𝛹; 𝐼(𝐷)) ) 𝜓𝑘−1 ( 𝑘 ) 𝑘 𝑘

∗ 𝜌𝑛−1 𝜌∗ (𝛹; 𝐼(𝐵)) (𝛷; 𝐼(𝐴)) −1 ) 𝜓𝑛−1 ) ( 𝑛−1 𝑛−1 𝑛−1

≤ 𝜙1−1 (𝜌1∗ (𝛷; 𝐼(𝐴)))𝜓1−1 (𝜌1∗ (𝛹; 𝐼(𝐵))) 𝑛−1

+ ∑ 𝜙𝑘−1 ( 𝑘=1

𝜌𝑘∗ (𝛷; 𝐼(𝐴)) 𝜌∗ (𝛹; 𝐼(𝐵)) ) 𝜓𝑘−1 ( 𝑘 ) 𝑘 𝑘

which is (4.95) for arbitrary 𝑛. Now, we examine Riemann–Stieltjes sums. As in Definition 4.4, we use the notation (4.17), where 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) is a partition of [𝑎, 𝑏] and 𝛱 = {𝜏1 , 𝜏2 , . . . , 𝜏𝑚 } is a set of intermediate points satisfying 𝑡𝑗−1 ≤ 𝜏𝑗 ≤ 𝑡𝑗 for 𝑗 = 1, 2, . . . , 𝑚. In the follow­ ing proposition, we give an upper estimate for (4.17) in terms of finitely many elements of two Schramm sequences whose number depends on the length 𝑚 of the partition 𝑃.

308 | 4 Riemann–Stieltjes integrals Proposition 4.39. Let 𝛷 = (𝜙𝑛 )𝑛 and 𝛹 = (𝜓𝑛 )𝑛 be two Schramm sequences, and fix sets 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]) and 𝛱 = {𝜏1 , 𝜏2 , . . . , 𝜏𝑚 } as above. Suppose that 𝑓 ∈ 𝛷𝐵𝑉𝑜 ([𝑎, 𝑏]) and 𝛼 ∈ 𝛹𝐵𝑉([𝑎, 𝑏]). Then the estimate 𝑚−1

𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) ≤ 2 ∑ 𝜙𝑗−1 ( 𝑗=1

Var𝛷 (𝑓) Var𝛹 (𝛼) ) 𝜓𝑗−1 ( ) 𝑗 𝑗

(4.97)

is true, where 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) is defined by (4.17). Proof. We prove the assertion by making special choices for the intermediate points in 𝛱. If we take 𝜏0 := 𝑎, we get 󵄨󵄨 𝑚 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 |𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏])| = 󵄨󵄨󵄨 ∑ 𝑓(𝜏𝑗 )(𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 ))󵄨󵄨󵄨󵄨 󵄨󵄨𝑗=1 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 𝑚 𝑗 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨󵄨 ∑ ∑(𝑓(𝜏𝑖 ) − 𝑓(𝜏𝑖−1 ))(𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )󵄨󵄨󵄨󵄨 . 󵄨󵄨 󵄨󵄨𝑗=1 𝑖=1 󵄨 󵄨

(4.98)

Now, we define a sequence 𝐴 = (𝑎𝑛 )𝑛 by 𝑎𝑛 := 𝑓(𝜏𝑛 ) − 𝑓(𝜏𝑛−1 ) for 𝑛 ≤ 𝑚 and 𝑎𝑛 := 0 otherwise, and another sequence 𝐵 = (𝑏𝑛 )𝑛 by 𝑏𝑛 := 𝛼(𝑡𝑛 ) − 𝛼(𝑡𝑛−1 ) for 𝑛 ≤ 𝑚 and 𝑏𝑛 := 0 otherwise. Then from (4.98), we have 󵄨󵄨󵄨 󵄨󵄨󵄨 𝑚 𝑗 󵄨 󵄨 |𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏])| = 󵄨󵄨󵄨󵄨∑ ∑ 𝑎𝑗 𝑏𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨𝑗=1 𝑖=1 󵄨 󵄨 −1 ∗ ≤ 𝜙1 (𝜌1 (𝛷; 𝐼(𝐴))) 𝜓1−1 (𝜌1∗ (𝛹; 𝐼(𝐵))) 𝑚−1

𝜌𝑗∗ (𝛷; 𝐼(𝐴))

𝑗=1

𝑗

+ ∑ 𝜙𝑗−1 (

) 𝜓𝑗−1 (

𝜌𝑗∗ (𝛹; 𝐼(𝐵)) 𝑗

)

≤ 𝜙1−1 (Var𝛷 (𝑓)) 𝜓1−1 (Var𝛹 (𝛼)) 𝑚−1

+ ∑ 𝜙𝑗−1 ( 𝑗=1

𝑚−1

≤ 2 ∑ 𝜙𝑗−1 ( 𝑗=1

Var𝛷 (𝑓) Var𝛹 (𝛼) ) 𝜓𝑗−1 ( ) 𝑗 𝑗

Var𝛷 (𝑓) Var𝛹 (𝛼) ) 𝜓𝑗−1 ( ) 𝑗 𝑗

which is (4.97). Now, we are in a position to state our main result on the existence of the RS integral (4.7) for 𝛼 in a larger class than just 𝐵𝑉([𝑎, 𝑏]). Theorem 4.40. Let 𝛷 = (𝜙𝑛 )𝑛 and 𝛹 = (𝜓𝑛 )𝑛 be two Schramm sequences satisfying (4.83). If 𝑓 ∈ 𝛷𝐵𝑉𝑜 ([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]) and 𝛼 ∈ 𝛹𝐵𝑉([𝑎, 𝑏]), then the Riemann–Stieltjes integral (4.7) exists. Proof. Let 𝑃 = {𝑥0 , 𝑥1 , . . . , 𝑥𝑚 } ∈ P([𝑎, 𝑏]) and 𝑄 = {𝑦0 , 𝑦1 , . . . , 𝑦𝑛 } ∈ P([𝑎, 𝑏]) be parti­ tions of [𝑎, 𝑏], and let 𝛯 = {𝜉1 , 𝜉2 , . . . , 𝜉𝑚 } and 𝐻 = {𝜂1 , 𝜂2 , . . . , 𝜂𝑛 }, respectively, be sets

4.4 Nonclassical RS-integrals

| 309

of corresponding intermediate points, i.e. 𝑥0 ≤ 𝜉1 ≤ 𝑥1 ≤ . . . ≤ 𝑥𝑚−1 ≤ 𝜉𝑚 ≤ 𝑥𝑚 ,

𝑦0 ≤ 𝜂1 ≤ 𝑦1 ≤ . . . ≤ 𝑦𝑛−1 ≤ 𝜂𝑛 ≤ 𝑦𝑛 .

We define two step functions 𝑔, ℎ : [𝑎, 𝑏] → ℝ by {0 𝑔(𝑡) := { 𝑓(𝜉 ) { 𝑗

for 𝑡 = 𝑎 , for 𝑥𝑖−1 < 𝑡 ≤ 𝑥𝑗

for 𝑖 = 1, 2, . . . , 𝑚 and {0 for 𝑡 = 𝑎 , ℎ(𝑡) := { 𝑓(𝜂 ) for 𝑦𝑗−1 < 𝑡 ≤ 𝑦𝑗 { 𝑗 for 𝑗 = 1, 2, . . . , 𝑛. These functions satisfy 𝑚

𝑚+𝑛

𝑆𝛼 (𝑔, 𝑃, 𝛯; [𝑎, 𝑏]) = ∑ 𝑔(𝜉𝑗 )(𝛼(𝑥𝑗 ) − 𝛼(𝑥𝑖−1 )) = ∑ 𝑔(𝑡𝑗 )(𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑖−1 )) 𝑖=1

𝑖=1

𝑛

𝑚+𝑛

and 𝑆𝛼 (ℎ, 𝑄, 𝐻; [𝑎, 𝑏]) = ∑ ℎ(𝜂𝑗 )(𝛼(𝑦𝑗 ) − 𝛼(𝑦𝑗−1 )) = ∑ ℎ(𝑡𝑗 )(𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )) , 𝑗=1

𝑗=1

respectively. Consequently , 𝑚+𝑛

𝑆𝛼 (𝑔, 𝑃, 𝛯; [𝑎, 𝑏]) − 𝑆𝛼 (ℎ, 𝑄, 𝐻; [𝑎, 𝑏]) = 2 ∑ 𝑘=1

1 (𝑔(𝑡𝑘 ) − ℎ(𝑡𝑘 )) (𝛼(𝑡𝑘 ) − 𝛼(𝑡𝑘−1 )) , (4.99) 2

the factors 2 and 1/2 being for later convenience. By 𝑅 = 𝑃 ∪ 𝑄 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚+𝑛 }, we denote the common refinement of the partitions 𝑃 and 𝑄. Then (4.99) may be rewritten in the form 𝑆𝛼 (𝑔, 𝑃, 𝛯; [𝑎, 𝑏]) − 𝑆𝛼 (ℎ, 𝑄, 𝐻; [𝑎, 𝑏]) = 2𝑆𝛼 (𝑑, 𝑅, 𝑅; [𝑎, 𝑏]) , where the function 𝑑 := 12 (𝑔 − ℎ) belongs to 𝛷𝐵𝑉𝑜 ([𝑎, 𝑏]) ⊆ 𝛤𝛷𝐵𝑉𝑜 ([𝑎, 𝑏]). Now, we apply Proposition 4.39 to the Schramm sequences 𝛤 ∘ 𝛷 and 𝛤 ∘ 𝛹, with 𝛤 as in Lemma 4.36 (b), and to 𝑓 replaced by 𝑑. As a result, we obtain the estimate 󵄨 󵄨󵄨 󵄨󵄨𝑆𝛼 (𝑔, 𝑃, 𝛯; [𝑎, 𝑏]) − 𝑆𝛼 (ℎ, 𝑄, 𝐻; [𝑎, 𝑏])󵄨󵄨󵄨 𝑚+𝑛

≤ 4 ∑ (𝛤 ∘ 𝜙𝑘 )−1 ( 𝑘=1

Var𝛤∘𝛷 (ℎ) Var𝛤∘𝛹 (𝛼) ) (𝛤 ∘ 𝜓𝑘 )−1 ( ). 𝑘 𝑘

We further get, with 𝜒𝑛 given by (4.87) and osc(𝑓; [𝑎, 𝑏]) given by (1.12), Var𝛤∘𝛷 (𝑑; [𝑎, 𝑏]) ≤ 𝜒1 (𝛷, osc(𝑑; [𝑎, 𝑏])) Var𝛷 (𝑑; [𝑎, 𝑏]) 1 ≤ 𝜒1 (𝛷, osc(𝑑; [𝑎, 𝑏])) (Var𝛷 (𝑔; [𝑎, 𝑏]) + Var𝛷 (ℎ; [𝑎, 𝑏])) 2 ≤ 𝜒1 (𝛷, osc(𝑑; [𝑎, 𝑏])) Var𝛷 (𝑓; [𝑎, 𝑏]) ≤ 𝜒1 (𝛷, osc((𝑔 − 𝑓)/2; [𝑎, 𝑏])) + osc((ℎ − 𝑓)/2; [𝑎, 𝑏])) Var𝛷 (𝑓; [𝑎, 𝑏]) .

310 | 4 Riemann–Stieltjes integrals Let 𝜀 > 0. Since 𝑓 is uniformly continuous on [𝑎, 𝑏], we may find 𝛿 > 0 such that osc((𝑔 − 𝑓)/2; [𝑎, 𝑏]) + osc((ℎ − 𝑓)/2; [𝑎, 𝑏]) ≤ 𝜀 for 𝜇(𝑃) ≤ 𝛿 and 𝜇(𝑄) ≤ 𝛿. Then 󵄨 󵄨󵄨 󵄨󵄨𝑆𝛼 (𝑔, 𝑃, 𝛯; [𝑎, 𝑏]) − 𝑆𝛼 (ℎ, 𝑄, 𝐻; [𝑎, 𝑏])󵄨󵄨󵄨 ∞ Var𝛤∘𝛹(𝑓) (𝛼) 𝜒 (𝛷; 𝜀) Var𝛷 (𝑓) ≤ ∑ (𝛤 ∘ 𝜙𝑘 )−1 ( 1 ) (𝛤 ∘ 𝜓𝑘 )−1 ( ). 𝑘 𝑘 𝑘=1

(4.100)

Since 𝜒1 (𝛷; 𝜀) → 0 as 𝜀 → 0, we may make the right-hand side of (4.100) as small as we wish by choosing 𝜀 sufficiently small. The proof is complete. When applying Theorem 4.40 to specific Schramm sequences 𝛷 = (𝜙𝑛 )𝑛 and 𝛹 = (𝜓𝑛 )𝑛 , the crucial condition to be verified is (4.83). As mentioned before, this condition triv­ ially holds if only finitely many terms 𝜙𝑛 and 𝜓𝑛 are different from zero. For example, in the most important example of the space 𝐵𝑉([𝑎, 𝑏]), this is true, and so we regain Theorem 4.11 as a special case of Theorem 4.40. A more interesting case, where the se­ ries (4.83) is infinite, refers to the Waterman space Λ𝐵𝑉([𝑎, 𝑏]) introduced in Definition 2.15. Let Λ = (𝜆 𝑛 )𝑛 and 𝑀 = (𝜇𝑛 )𝑛 be two Waterman sequences in the sense of Defini­ tion 2.15. Then the corresponding spaces Λ𝐵𝑉([𝑎, 𝑏]) and 𝑀𝐵𝑉([𝑎, 𝑏]), equipped with the norms (2.30), may be interpreted as Schramm spaces 𝛷𝐵𝑉([𝑎, 𝑏]) and 𝛹𝐵𝑉([𝑎, 𝑏]) by choosing 𝜙𝑛 (𝑡) = 𝜆 𝑛𝑡 and 𝜓𝑛 (𝑡) = 𝜇𝑛 𝑡. Consequently, the crucial condition (4.83) holds if and only if ∞ 1 ∑ 2 < ∞. (4.101) 𝑘 𝜆 𝑘 𝜇𝑘 𝑘=1 Therefore, as a corollary of Theorem 4.40, we obtain the following Theorem 4.41. Let Λ = (𝜆 𝑛 )𝑛 and 𝑀 = (𝜇𝑛 )𝑛 be two Waterman sequences satisfying (4.101). Then the Riemann–Stieltjes integral (4.7) exists for 𝑓 ∈ Λ𝐵𝑉𝑜 ([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]) and 𝛼 ∈ 𝑀𝐵𝑉([𝑎, 𝑏]). Moreover, the estimate 󵄨󵄨 𝑏 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨 󵄨󵄨∫ 𝑓(𝑥) 𝑑𝛼(𝑥)󵄨󵄨󵄨 ≤ 2‖𝑓‖Λ𝐵𝑉 ‖𝛼‖Λ𝐵𝑉 ∑ 2 󵄨󵄨 󵄨󵄨 𝑘 𝜆 𝑘 𝜇𝑘 𝑘=1 󵄨󵄨 𝑎 󵄨󵄨

(4.102)

holds, where ‖ ⋅ ‖Λ𝐵𝑉 denotes the norm (2.30). We illustrate Theorem 4.41 by a simple example where condition (4.101) may be easily verified. Example 4.42. Choosing 𝜆 𝑛 := 𝑛−𝑝 and 𝜇𝑛 := 𝑛−𝑞 for 0 < 𝑝, 𝑞 ≤ 1, we get the Waterman spaces Λ 𝑝 𝐵𝑉([𝑎, 𝑏]) and Λ 𝑞 𝐵𝑉([𝑎, 𝑏]) which we introduced in Definition 2.29. For this choice of 𝜆 𝑛 and 𝜇𝑛 , condition (4.101) reads ∞

1 = 𝜁(2 − 𝑝 − 𝑞, 0) < ∞ , 2−𝑝−𝑞 𝑘 𝑘=1 ∑

4.5 Comments on Chapter 4 | 311

where we have used the shortcut (0.17), and this is true precisely for 𝑝 + 𝑞 < 1. So, in this case, we get the estimate 󵄨󵄨 󵄨󵄨 𝑏 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨∫ 𝑓(𝑥) 𝑑𝛼(𝑥)󵄨󵄨󵄨 ≤ 2𝜁(2 − 𝑝 − 𝑞, 0) ‖𝑓‖Λ 𝑝 𝐵𝑉 ‖𝛼‖Λ𝐵𝑉𝑞 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑎 ♥

for any pair of functions 𝑓 ∈ Λ 𝑝 𝐵𝑉([𝑎, 𝑏]) and 𝛼 ∈ Λ 𝑞 𝐵𝑉([𝑎, 𝑏]).

4.5 Comments on Chapter 4 Riemann–Stieltjes integrals are treated in many calculus textbooks, e.g. [76, 139, 146– 148, 156, 181, 186, 238]. In Section 4.1, we followed the presentation in [269] and [182], and in the other section, as in Chapter 3, in Carothers’ beautiful book [76]. The particularly important Theorem 4.11 shows that 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) implies 𝐶([𝑎, 𝑏]) ⊆ 𝑅𝑆𝛼 ([𝑎, 𝑏]), while 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]) implies 𝐵𝑉([𝑎, 𝑏]) ⊆ 𝑅𝑆𝛼 ([𝑎, 𝑏]). In particular, continuous functions and functions of bounded variation are Riemann integrable. When passing from increasing integrators to 𝐵𝑉-integrators, we used for 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) the decomposition 𝛼 = 𝛽 − 𝛾 and the fact that 𝑅𝑆𝛽 ([𝑎, 𝑏]) ∩ 𝑅𝑆𝛾 ([𝑎, 𝑏]) ⊆ 𝑅𝑆𝛽−𝛾 ([𝑎, 𝑏]) = 𝑅𝑆𝛼 ([𝑎, 𝑏]) ,

(4.103)

by Proposition 4.3 (b). We would of course like to have equality in (4.103), i.e. (4.104)

𝑅𝑆𝛽 ([𝑎, 𝑏]) ∩ 𝑅𝑆𝛾 ([𝑎, 𝑏]) = 𝑅𝑆𝛼 ([𝑎, 𝑏])

because this would truly reduce the study of 𝐵𝑉-integrators to the case of increasing integrators. Unfortunately, (4.104) is generally not true for just any splitting 𝛼 = 𝛽 − 𝛾. For example, in case 𝛽(𝑥) = 𝛾(𝑥) = 𝑥, we have 𝛼(𝑥) ≡ 0, and so the right-hand side of (4.103) is much larger than just 𝑅𝑆([𝑎, 𝑏]). However, one can show that (4.104) is true for the canonical Jordan decomposition from Theorem 1.5, see Exercise 4.41. Theorems 4.14 and 4.15 are crucial for the existence of the Riemann–Stieltjes inte­ gral. Given a partition 𝑃 ∈ P([𝑎, 𝑏]) and using the notation (4.2) and (4.3), we have the estimate 𝑈𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) − 𝐿 𝛼 (𝑓, 𝑃; [𝑎, 𝑏]) ≥ osc(𝑓; 𝑥) osc(𝛼; 𝑥)

(𝑥 ∈ ̸ 𝑃) ,

where osc(𝑓; 𝑐) denotes the local oscillation (4.27). Consequently, in order for 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]), we must have osc(𝑓; 𝑥) osc(𝛼; 𝑥) = 0 for “most” values of 𝑥, in accordance with Theorem 4.15. We point out that the convergence relation (4.47) holds if we only suppose that (𝛼𝑛 )𝑛 is a bounded sequence in 𝐵𝑉([𝑎, 𝑏]) which converges pointwise to 𝛼. This fact is known as Helly’s first theorem. Example 4.22 is taken from Rudin’s book [269]. Theorem 4.17 gives conditions under which a Riemann–Stieltjes integral reduces to a Riemann integral. What about Lebesgue integrals? We take this one step further and now consider the right integral in (4.34) as a Lebesgue integral.

312 | 4 Riemann–Stieltjes integrals Theorem 4.43. Let 𝛼 ∈ 𝐴𝐶([𝑎, 𝑏]) and 𝑓 ∈ 𝐶([𝑎, 𝑏]). Then 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) and 𝑏

𝑏

(4.105)

̇ 𝑑𝑡, ∫ 𝑓(𝑡) 𝑑𝛼(𝑡) = ∫ 𝑓(𝑡)𝛼(𝑡) 𝑎

𝑎

where 𝛼̇ denotes the derivative of 𝛼 with respect to 𝑡 a.e. on [𝑎, 𝑏], and the integral on the right-hand side of (4.105) is the Lebesgue integral. Proof. Given a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } and a set 𝛱 = {𝜏1 , 𝜏2 , . . . , 𝜏𝑚 } of intermedi­ ate points satisfying (4.16), consider the Riemann–Stieltjes sum (4.17). Since 𝛼 is ab­ solutely continuous, we may write 𝑡𝑗

̇ 𝑑𝑡, 𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 ) = ∫ 𝛼(𝑡) 𝑡𝑗−1

and hence 𝑚

𝑡𝑗

𝑚

̇ 𝑑𝑡 . 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) = ∑ 𝑓(𝜏𝑗 )[𝛼(𝑡𝑗 ) − 𝛼(𝑡𝑗−1 )] = ∑ ∫ 𝛼(𝑡) 𝑗=1

𝑗=1 𝑡

𝑗−1

Consequently, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 𝑚 𝑡𝑗 𝑏 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 ̇ ̇ 𝑆 (𝑓, 𝑃.𝛱; [𝑎, 𝑏]) − ∫ 𝑓(𝑡) 𝛼(𝑡) 𝑑𝑡 = ∑ ∫ [𝑓(𝜏 ) − 𝑓(𝑡)] 𝛼(𝑡) 𝑑𝑡 󵄨󵄨 󵄨󵄨 𝑗 󵄨󵄨󵄨 𝛼 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑗=1 󵄨󵄨 󵄨󵄨 󵄨 󵄨 𝑎 󵄨 󵄨 󵄨󵄨 𝑡𝑗−1 󵄨󵄨 𝑡𝑗

𝑚

̇ 𝑑𝑡 ≤ ∑ osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) ∫ |𝛼(𝑡)| 𝑗=1

𝑡𝑗−1

≤ ‖𝛼‖̇ 𝐿 1 max {osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) : 𝑗 = 1, 2, . . . , 𝑚} . Since 𝑓 is continuous, by assumption, we have osc(𝑓; [𝑡𝑗−1 , 𝑡𝑗 ]) → 0 as 𝜇(𝑃) → 0, and so 𝑏

̇ 𝑑𝑡 lim 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) = ∫ 𝑓(𝑡)𝛼(𝑡)

𝜇(𝑃)→0

𝑎

which proves the assertion. As an immediate consequence of Theorem 4.43, we get the following result which is parallel to Proposition 4.24: Proposition 4.44 (integration by parts). Let 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) and 𝛼 ∈ 𝐴𝐶([𝑎, 𝑏]). Then the equality 𝑏

𝑏

∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = 𝑓(𝑏)𝛼(𝑏) − 𝑓(𝑎)𝛼(𝑎) − ∫ 𝛼(𝑥) 𝑑𝑓(𝑥) 𝑎

holds.

𝑎

(4.106)

4.5 Comments on Chapter 4

| 313

One of the most important properties of the Riemann–Stieltjes integral is that of pro­ viding the right tool for the duality theory exposed in Sections 4.2 and 4.3. Our proof of Theorem 4.31 is purely “functional-analytic,” being based on the Hahn–Banach the­ orem, and has the advantage of carrying over with only minor technical changes to other function spaces. However, it is interesting to note that there exist other proofs which are purely “analytic” and essentially use special properties of the space 𝐶([𝑎, 𝑏]). We give here a proof taken from the book [76] of the surjectivity of the map 𝛷 which is the crucial part of Theorem 4.31. Without loss of generality, let [𝑎, 𝑏] = [0, 1]. The proof in [76] uses the Bernstein polynomials 𝑝𝑛,𝑘 : [0, 1] → ℝ (𝑛 = 1, 2, 3, . . .) defined by 𝑛 𝑝𝑘,𝑛 (𝑥) := ( )𝑥𝑘 (1 − 𝑥)𝑛−𝑘 (𝑘 = 0, 1, 2, . . . , 𝑛) . 𝑘 It is well known that for any function 𝑓 ∈ 𝐶([0, 1]), the sequence (𝐵𝑛 (𝑓))𝑛 given by 𝑛

𝐵𝑛(𝑓) := ∑ 𝑓(𝑘/𝑛)𝑝𝑘,𝑛 𝑘=0

converges uniformly on [0, 1] to 𝑓 as 𝑛 → ∞. We use this fact to now give an alternative proof of the following part of Theorem 4.31: Theorem 4.45. The map 𝛷 : 𝐵𝑉([𝑎, 𝑏]) → 𝐶([𝑎, 𝑏])∗ ,

𝛷(𝛼) := ℓ𝛼 ,

with ℓ𝛼 as in (4.61) is onto. Proof. Let ℓ be an arbitrary bounded linear functional on 𝐶([0, 1]). Then for any 𝑓 ∈ 𝐶([0, 1]), we have 𝑛

ℓ(𝐵𝑛 (𝑓)) = ∑ 𝑓(𝑘/𝑛)ℓ(𝑝𝑘,𝑛 ) → ℓ(𝑓)

(𝑛 → ∞)

𝑘=0

by what we have observed before. In particular, observe that the numbers ℓ(𝑝𝑘,𝑛 ) (𝑛 = 1, 2, 3, . . . ; 𝑘 = 0, 1, 2, . . . , 𝑛) do not depend on 𝑓. We define a sequence (𝛼𝑛 )𝑛 of functions 𝛼𝑛 ∈ 𝐵𝑉([0, 1]) by¹⁴ 0 { { { { { {ℓ(𝑝𝑛,1 ) 𝛼𝑛 (𝑥) := { { ℓ(𝑝𝑛,𝑘 ) { { { { {ℓ(𝑝𝑛,𝑛 )

for 𝑥 = 0 , for 0 < 𝑥 < for

𝑘 𝑛

≤𝑥
0, there exists a 𝛿 > 0 such that |𝑆2𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) − 𝐴| ≤ 𝜀 for any partition 𝑃 satisfying 𝜇(𝑃) ≤ 𝛿 and any set 𝛱 of intermediate points satisfying (4.16). Then we write 𝑓 ∈ 𝑅𝑆2𝛼 ([𝑎, 𝑏]) and call the number 𝑏

𝐴 = ∫ 𝑓(𝑥) 𝑎

𝑑2 𝛼(𝑥) 𝑑𝑥

the second order Riemann–Stieltjes integral of 𝑓 with respect to 𝛼.

◼

In the papers [272, 275, 277], one can find some natural properties of this integral as well as a connection with the usual RS-integral (4.7). Suppose, for example, that 𝑓 ∈ 𝐶([𝑎, 𝑏]) and 𝛽 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]), and let 𝑥

𝛼(𝑥) := ∫ 𝛽(𝑡) 𝑑𝑡

(𝑎 ≤ 𝑥 ≤ 𝑏) .

𝑎

Then 𝛼 ∈ 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]) and, as one should expect , 𝑏

𝑏

𝑎

𝑎

𝑑2 𝛼(𝑥) ∫ 𝑓(𝑥) = ∫ 𝑓(𝑥)𝑑𝛽(𝑥) . 𝑑𝑥 Riemann–Stieltjes integrals have been extended in many different directions. We confine ourselves to mention the paper [248] where the author defines generalized RSintegrals with respect to charges¹⁶ and studies the linear space of all functions which are integrable with respect to every charge. Related questions are discussed in [150].

15 In fact, Russell’s definition is much more general, involving another increasing function 𝑢 and using terms like 𝑢-bounded variation, 𝑢-derivative, 𝑢-convexity etc.; our definition corresponds to the special case 𝑢(𝑥) = 𝑥. 16 In the terminology of [248], a charge is an additive continuous function on bounded subsets of 𝐵𝑉.

316 | 4 Riemann–Stieltjes integrals

4.6 Exercises to Chapter 4 We state now some exercises on the topics covered in this chapter; exercises marked with an asterisk * are more difficult. Exercise 4.1. Calculate the integral 𝜋

∫ (𝑥 + 2) 𝑑𝛼(𝑥) , −𝜋

where 𝛼(𝑥) = 𝑒𝑥 [𝜒(0,1] − 𝜒[−1,0)] ]. Exercise 4.2. Calculate the integral 𝜋

∫(𝑥 − 1) 𝑑𝛼(𝑥) , 0

where 𝛼(𝑥) = cos 𝑥 𝑠𝑔𝑛 𝑥. Exercise 4.3. Calculate the integral 1

∫ 𝑥 𝑑𝜑(𝑥) , 0

where 𝜑 denotes the Cantor function (3.6) on [0, 1]. Exercise 4.4. Calculate the integral 1

∫ 𝑥3 𝑑𝜑(𝑥) , 0

where 𝜑 denotes the Cantor function (3.6) on [0, 1]. Exercise 4.5. Show that

1

∫ 𝜑(1 − 𝑥) 𝑑𝜑(𝑥) = 0

1 , 2

where 𝜑 denotes the Cantor function (3.6) on [0, 1]. Exercise 4.6. Give an example of functions 𝑓 ∈ 𝐵([𝑎, 𝑏]) and 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) such that 𝑓2 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) and |𝑓| ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]), but 𝑓 ∈ ̸ 𝑅𝑆𝛼 ([𝑎, 𝑏]). Exercise 4.7. Prove that 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) implies 𝑓3 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]). Is the converse also true? Exercise 4.8. Find functions 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) with 𝑓([𝑎, 𝑏]) ⊆ [𝑐, 𝑑] and 𝑔 ∈ 𝑅𝑆𝛼 ([𝑐, 𝑑]) such that 𝑔 ∘ 𝑓 ∈ ̸ 𝑅𝑆𝛼 ([𝑎, 𝑏]).

4.6 Exercises to Chapter 4 | 317

Exercise 4.9. Find functions 𝑓 ∈ 𝐶([𝑎, 𝑏]) with 𝑓([𝑎, 𝑏]) ⊆ [𝑐, 𝑑] and 𝑔 ∈ 𝑅𝑆𝛼 ([𝑐, 𝑑]) such that 𝑔 ∘ 𝑓 ∈ ̸ 𝑅𝑆𝛼 ([𝑎, 𝑏]). Exercise 4.10. Let 𝑓 : [−1, 1] → ℝ be bounded. We define three functions 𝛼𝑖 ∈ 𝐵𝑉([−1, 1]) (𝑖 = 1, 2, 3) by {0 𝛼1 (𝑡) := { 2 {

for − 1 ≤ 𝑡 ≤ 0 , for 0 < 𝑡 ≤ 1 ,

{0 for − 1 ≤ 𝑡 < 0 , 𝛼2 (𝑡) := { 2 for 0 ≤ 𝑡 ≤ 1 , {

and 0 { { { 𝛼3 (𝑡) := 𝛼1 (𝑡) + 𝛼2 (𝑡) = {1 { { {2

for − 1 ≤ 𝑡 < 0 , for 𝑡 = 0 , for 0 < 𝑡 ≤ 1 .

(a) Show that 𝑓 ∈ 𝑅𝑆𝛼1 ([−1, 1]) if and only if 𝑓 is right continuous at zero, i.e. 𝑓(0+) = 𝑓(0); in this case, we have 1

∫ 𝑓(𝑥) 𝑑𝛼1 (𝑥) = 𝑓(0) . −1

(b) Formulate and prove an analogous result for 𝛼2 . (c) Show that 𝑓 ∈ 𝑅𝑆𝛼3 ([−1, 1]) if and only if 𝑓 is continuous at zero. (d) Deduce that if 𝑓 is continuous at zero, the equality 1

1

1

∫ 𝑓(𝑥) 𝑑𝛼1 (𝑥) = ∫ 𝑓(𝑥) 𝑑𝛼2 (𝑥) = ∫ 𝑓(𝑥) 𝑑𝛼3 (𝑥) = 𝑓(0) −1

−1

−1

is true. Exercise 4.11. Let (𝑥𝑛 )𝑛 be a sequence of pairwise different real numbers 𝑥𝑛 ∈ (0, 1), and let (𝑐𝑛 )𝑛 be a positive real sequence such that ∞

∑ 𝑐𝑛 < ∞ .

𝑛=1

Define 𝛼 : [−1, 1] → ℝ by ∞

𝛼(𝑥) := ∑ 𝑐𝑛 𝛼1 (𝑥 − 𝑥𝑛 ) , 𝑛=1

where 𝛼1 is defined as in the preceding Exercise 4.10. Show that 𝑓 ∈ 𝑅𝑆𝛼 ([−1, 1]) for any 𝑓 ∈ 𝐶([−1, 1]) and 1

∞

∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = ∑ 𝑐𝑛 𝑓(𝑥𝑛 ) . −1

𝑛=1

318 | 4 Riemann–Stieltjes integrals Exercise 4.12. Show that

3

∫ 𝑥 𝑑𝛼(𝑥) = 0

3 , 2

where 𝛼(𝑥) := 𝑥 − ent 𝑥. Exercise 4.13. Construct functions 𝑓 ∈ 𝐶([−1, 1]) and 𝛼 ∈ 𝐵𝑉([−1, 1]) such that 𝑓𝜒[0,1] ∈ ̸ 𝑅𝑆𝛼 ([−1, 1]). Exercise 4.14. Let 𝑐 ∈ (𝑎, 𝑏) and 𝛼 := 𝜒[𝑐,𝑏] , and let 𝑓 : [𝑎, 𝑏] → ℝ be some function. Prove that 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) if and only if 𝑓 is continuous at 𝑐. What is the value of the RS-integral 𝑏

𝐼 = ∫ 𝑓(𝑥) 𝑑𝛼(𝑥) 𝑎

in this case? Exercise 4.15. For 𝜎 > 0 and 𝜏 ∈ ℝ, let 𝑓𝜎 : [0, 1] → ℝ and 𝛼𝜏 : [0, 1] → ℝ be defined by {𝑥𝜎 for 0 < 𝑥 ≤ 1 , {𝑥𝜏 for 0 < 𝑥 ≤ 1 , 𝛼𝜏 (𝑥) := { 𝑓𝜎 (𝑥) := { 0 for 𝑥 = 0 , 0 for 𝑥 = 0 . { { Show that 𝑓𝜎 ∈ 𝑅𝑆𝛼𝜏 ([0, 1]) if and only if 𝜏 ≥ 0, and calculate the RS-integral of 𝑓𝜎 with respect to 𝛼𝜏 in this case. Exercise 4.16. For 𝜎 ∈ ℝ, let 𝑓𝜎 : [0, 1] → ℝ and 𝛼 : [0, 1] → ℝ be defined by {cos 𝑥1 for 0 < 𝑥 ≤ 1 , {𝑥𝜎 for 0 < 𝑥 ≤ 1 , 𝛼(𝑥) := { 𝑓𝜎 (𝑥) := { 0 for 𝑥 = 0 , 0 for 𝑥 = 0 . { { Show that 𝑓𝜎 ∈ 𝑅𝑆𝛼 ([0, 1]) if and only if 𝜎 > 0, and calculate the RS-integral of 𝑓𝜎 with respect to 𝛼 in this case. Exercise 4.17. Let 𝑓 = 𝜒[0,1]∩ℚ be the Dirichlet function, and let 𝛼 ∈ 𝐵𝑉([0, 1]). Show that 𝑓 ∈ 𝑅𝑆𝛼 ([0, 1]) if and only if 𝛼 is constant. Exercise 4.18. Let 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), and suppose that 𝑓 ∈ 𝐵([𝑎, 𝑏]) belongs to both 𝑅𝑆𝛼 ([𝑎, 𝑐]) and 𝑅𝑆𝛼 ([𝑐, 𝑏]) for some 𝑐 ∈ (𝑎, 𝑏). Assume that 𝛼 is continuous at 𝑐. Prove that then 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) with (4.23). Exercise 4.19. Let 𝑓, 𝑔 ∈ 𝐶([𝑎, 𝑏]) and 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), and define 𝛽 : [𝑎, 𝑏] → ℝ by 𝑥

𝛽(𝑥) := ∫ 𝑓(𝑡) 𝑑𝛼(𝑡)

(𝑎 ≤ 𝑥 ≤ 𝑏) .

𝑎

Show that

𝑏

𝑏

∫ 𝑔(𝑡) 𝑑𝛽(𝑡) = ∫ 𝑓(𝑡)𝑔(𝑡) 𝑑𝛼(𝑡) 𝑎

𝑎

and illustrate this by means of a nontrivial example.

4.6 Exercises to Chapter 4

| 319

Exercise 4.20. Using the notation of Definition 4.4 and Exercise 4.10, show that 𝛼2 ∈ 𝑅𝑆𝛼1 ([−1, 1]), although the limit lim 𝑆𝛼1 (𝛼2 , 𝑃; [−1, 1])

𝜇(𝑃)→0

does not exist. Why does this not contradict Theorem 4.11 (b)? Exercise 4.21. Give an example of functions 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) and 𝑓 ∈ 𝐶([𝑎, 𝑏]) \ 𝐵𝑉([𝑎, 𝑏]) such that one may define the RS-integral 𝑏

∫ 𝛼(𝑥) 𝑑𝑓(𝑥) 𝑎

building on the equality (4.55) in Proposition 4.24. Exercise 4.22. Let 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]), and let 𝑓 : [𝑎, 𝑏] → ℝ be monotonically increasing. Prove that there exists 𝜉 ∈ [𝑎, 𝑏] such that 𝑏

∫ 𝑓(𝑥) 𝑑𝛼(𝑥) = 𝑓(𝑎)(𝛼(𝜉) − 𝛼(𝑎)) + 𝑓(𝑏)(𝛼(𝑏) − 𝛼(𝜉)) . 𝑎

Exercise 4.23. Given 𝑓 ∈ 𝐶([𝑎, 𝑏]), prove that there exists a point 𝜉 ∈ [𝑎, 𝑏] such that 𝑏

∫ 𝑓(𝑥) 𝑑𝑥 = 𝑓(𝜉)(𝑏 − 𝑎) . 𝑎

Exercise 4.24. Let 𝑓, 𝑔 ∈ 𝐶([𝑎, 𝑏]) with 𝑔(𝑥) ≥ 0 on [𝑎, 𝑏]. Prove that there exists a point 𝜉 ∈ [𝑎, 𝑏] such that 𝑏

𝑏

∫ 𝑓(𝑥)𝑔(𝑥) 𝑑𝑥 = 𝑓(𝜉) ∫ 𝑔(𝑥) 𝑑𝑥 . 𝑎

𝑎

Show that this result contains that of Exercise 4.23 as a special case, and that one cannot drop the assumption 𝑔(𝑥) ≥ 0. Exercise 4.25. Let 𝑓 ∈ 𝐶1 ([𝑎, 𝑏]) and 𝑔 ∈ 𝐶([𝑎, 𝑏]). Prove that there exists a point 𝑐 ∈ (𝑎, 𝑏) such that 𝑏

𝑐

𝑏

∫ 𝑓(𝑥)𝑔(𝑥) 𝑑𝑥 = 𝑓(𝑎) ∫ 𝑔(𝑥) 𝑑𝑥 + 𝑓(𝑏) ∫ 𝑔(𝑥) 𝑑𝑥 . 𝑎

𝑎

𝑐

Exercise 4.26. Let 𝑥0 ∈ [𝑎, 𝑏] be fixed. Construct three functions 𝑓 : [𝑎, 𝑏] → ℝ, 𝛼 : [𝑎, 𝑏] → ℝ, and 𝛽 : [𝑎, 𝑏] → ℝ such that 𝛥(𝛼, 𝛽) = {𝑥0 } (see (4.66)) and 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) \ 𝑅𝑆𝛽 ([𝑎, 𝑏]).

320 | 4 Riemann–Stieltjes integrals Exercise 4.27. Given a sequence (𝑥𝑛 )𝑛 of points 𝑥𝑛 ∈ [𝑎, 𝑏], define 𝛾 : [𝑎, 𝑏] → ℝ by ∞

𝛾(𝑡) := ∑ (𝑎𝑛 𝜒[𝑥𝑛 ,𝑏] (𝑡) + 𝑏𝑛 𝜒(𝑥𝑛 ,𝑏] (𝑡)) , 𝑛=1

where (𝑎𝑛 )𝑛 and (𝑏𝑛 )𝑛 are two real sequence satisfying ∞

∞

∑ |𝑎𝑛 | < ∞,

∑ |𝑏𝑛 | < ∞ .

𝑛=1

𝑛=1

(a) Show that 𝛾 ∈ 𝐵𝑉([𝑎, 𝑏]). (b) If 𝑓 : [𝑎, 𝑏] → ℝ is a bounded function which is continuous at each point 𝑥𝑛 , prove that 𝑓 ∈ 𝑅𝑆𝛾 ([𝑎, 𝑏]) and 𝑏

∞

∫ 𝑓(𝑥) 𝑑𝛾(𝑥) = ∑ (𝑎𝑛 + 𝑏𝑛 )𝑓(𝑥𝑛 ) . 𝑛=1

𝑎

Exercise 4.28. Let 𝛼𝑛 : [0, 1] → ℝ be defined by 𝛼𝑛 (𝑥) :=

ent (𝑛𝑥 ) 𝑛

(𝑛 = 1, 2, 3, . . .) ,

where ent(𝜉) denotes the integer part of 𝜉. Show that 𝛼𝑛 ∈ 𝐵𝑉([0, 1]) and that the se­ quence (𝛼𝑛 )𝑛 converges in the norm (1.12) to some 𝛼 ∈ 𝐵𝑉([0, 1]). Given 𝑓 ∈ 𝐶([0, 1]), use Theorem 4.21 (c) to show that lim

𝑛→∞

1 𝑛 ∑ 𝑓(log 𝑘/ log 𝑛) = 𝑓(1) . 𝑛 𝑘=2

Illustrate this result for the example 𝑓(𝑥) = 𝑥2 . Exercise 4.29. Let 𝛼 ∈ 𝐵𝑉([0, 2𝜋]) with 𝛼(0) = 𝛼(2𝜋). Show that then the estimate 󵄨󵄨󵄨 󵄨󵄨󵄨 2𝜋 󵄨󵄨 1 󵄨󵄨 󵄨󵄨󵄨 ∫ 𝛼(𝑥) sin 𝑛𝑥 𝑑𝑥󵄨󵄨󵄨 ≤ Var(𝛼; [0, 2𝜋]) 󵄨󵄨 𝑛 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 holds for all 𝑛 ∈ ℕ. Compare with Proposition 4.23 (b). ̂ Exercise 4.30. Show that there exists a function 𝛼̂ ∈ 𝐵𝑉([0, 2𝜋]) such that 𝛼(0) = ̂ 𝛼(2𝜋) and 󵄨󵄨 󵄨󵄨 2𝜋 󵄨󵄨 1 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ∫ 𝛼(𝑥) 󵄨󵄨 ̂ sin 𝑛𝑥 𝑑𝑥󵄨󵄨󵄨 = 𝑛 Var(𝛼;̂ [0, 2𝜋]) 󵄨󵄨 󵄨󵄨 󵄨 󵄨0 for infinitely many 𝑛 ∈ ℕ. Compare this with Proposition 4.23 (b) and Exercise 4.29. Exercise 4.31. Let 𝑓, 𝛼 : [𝑎, 𝑏] → ℝ be continuous and let 𝛼 be in addition strictly increasing. Define 𝐹 : [𝑎, 𝑏] → ℝ by 𝑥

𝐹(𝑥) := ∫ 𝑓(𝑡) 𝑑𝛼(𝑡) 𝑎

(𝑎 ≤ 𝑥 ≤ 𝑏) .

4.6 Exercises to Chapter 4 |

321

The so-called 𝛼-derivative of 𝐹 is then defined by 𝐹(𝑥 + ℎ) − 𝐹(𝑥) 𝑑𝐹 (𝑥) := lim . ℎ→0 𝛼(𝑥 + ℎ) − 𝛼(𝑥) 𝑑𝛼 Prove that

𝑑𝐹 (𝑥) = 𝑓(𝑥) 𝑑𝛼

for all 𝑥 ∈ [𝑎, 𝑏]. Exercise 4.32. Let 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]). Prove that 𝑏

∫ 𝑓(𝑥) 𝑑𝑓(𝑥) = 𝑎

𝑓2 (𝑏) − 𝑓2 (𝑎) . 2

Use this result to solve again Exercise 4.5. Exercise 4.33. Let 𝑓 ∈ 𝐶([1, ∞)), and let 𝛼(𝑡) := ent(𝑡) denote the integer part of 𝑡 ≥ 1. Compute the RS-integral 𝑥

𝐹(𝑥) := ∫ 𝑓(𝑡)𝑑𝛼(𝑡)

(𝑥 ≥ 1)

1

for 𝑥 ∈ ℕ and 𝑥 ∈ ̸ ℕ. Exercise 4.34. Suppose that 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) has the property that the space 𝑅𝑆𝛼 ([𝑎, 𝑏]) contains all step functions. Prove that 𝛼 ∈ 𝐶([𝑎, 𝑏]). Exercise 4.35. Show that ⋂ 𝑅𝑆𝛼 ([𝑎, 𝑏]) = 𝐶([𝑎, 𝑏]) , 𝛼↗

where the intersection is taken over all increasing functions 𝛼 : [𝑎, 𝑏] → ℝ. Exercise 4.36. If 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]), show that the value of the RS-integral (4.7) does not depend on the values of 𝑓 at any finite number of points. Is this still true if we change “finite” to “countable?” Explain. Exercise 4.37. Let 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), 𝑓 ∈ 𝐵([𝑎, 𝑏]), 𝑃 ∈ P([𝑎, 𝑏]), and 𝛱 a set of interme­ diate points. Prove that |𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏])| ≤ ‖𝑓‖∞ Var(𝛼, 𝑃; [𝑎, 𝑏]) , where 𝑆𝛼 (𝑓, 𝑃, 𝛱; [𝑎, 𝑏]) denotes the RS-sum (4.17), ‖𝑓‖∞ the norm (0.39) of 𝑓, and Var(𝛼, 𝑃; [𝑎, 𝑏]) the variation (1.3) of 𝛼. Exercise 4.38. Let 𝑓 ∈ 𝐶1 ([1, ∞)), and let 𝛼 be defined as in Exercise 4.33. Using the integration by parts formula (4.55), prove that 𝑛

𝑛

∑ 𝑓(𝑘) = 𝛼(𝑛)𝑓(𝑛) − ∫ 𝑓󸀠 (𝑡)𝛼(𝑡) 𝑑𝑡 𝑘=1

1

322 | 4 Riemann–Stieltjes integrals and

2𝑛

2𝑛

𝑘

∑ (−1) 𝑓(𝑘) = ∫ 𝑓󸀠 (𝑡)(𝛼(𝑡) − 2𝛼(𝑡/2) 𝑑𝑡 𝑘=1

1

for any 𝑛 ∈ ℕ. Exercise 4.39*. Suppose that 𝛼 : [𝑎, 𝑏] → ℝ is right-continuous and increasing. Given 𝜀 > 0 and a subinterval [𝑐, 𝑑] ⊂ [𝑎, 𝑏], construct a continuous function 𝑓 : [𝑎, 𝑏] → [0, 1] such that 𝑏

∫ 𝑓 𝑑𝛼 ≥ 𝛼(𝑑) − 𝛼(𝑐) − 𝜀 . 𝑎

Can you also take 𝜀 = 0? Exercise 4.40*. Suppose that 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) is right-continuous. Given 𝜀 > 0 and a partition 𝑃 ∈ P([𝑎, 𝑏]), construct a continuous function 𝑓 : [𝑎, 𝑏] → [0, 1] such that 𝑏

∫ 𝑓 𝑑𝛼 ≥ Var(𝛼, 𝑃; [𝑎, 𝑏]) − 𝜀 . 𝑎

Use this to again prove the equality (4.72). Exercise 4.41. Let 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), 𝛽(𝑥) := 𝑉𝛼 (𝑥) = Var(𝛼; [𝑎, 𝑥]), and 𝛾(𝑥) := 𝛽(𝑥)−𝛼(𝑥). Show that 𝑅𝑆𝛼 ([𝑎, 𝑏]) = 𝑅𝑆𝛽 ([𝑎, 𝑏]) ∩ 𝑅𝑆𝛾 ([𝑎, 𝑏]). Exercise 4.42. Find functions 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) and 𝑓 ∈ 𝑅𝑆𝛼 ([𝑎, 𝑏]) such that 󵄨󵄨 𝑏 󵄨󵄨 𝑏 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨∫ 𝑓(𝑥) 𝑑𝛼(𝑥)󵄨󵄨󵄨 > ∫ |𝑓(𝑥)| 𝑑𝛼(𝑥) . 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑎 󵄨󵄨 𝑎 Exercise 4.43. Let 𝜏 : [𝑎, 𝑏] → [𝑐, 𝑑] be strictly increasing, continuous, and surjective (and hence a homeomorphism). Given 𝑔 ∈ 𝑅𝑆𝛼 ([𝑐, 𝑑]), show that 𝑓 := 𝑔∘𝜏 ∈ 𝑅𝑆𝛽 ([𝑎, 𝑏]), where 𝛽 := 𝛼 ∘ 𝜏. Moreover, prove that 𝑏

𝑑

∫ 𝑓 𝑑𝛽 = ∫ 𝑔 𝑑𝛼 𝑎

𝑐

in this case. Exercise 4.44. Let 𝑝𝛼 and 𝑛𝛼 denote the positive respectively negative variation (The­ orem 1.6) of 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]), so 𝛼(𝑥) = 𝑝𝛼 (𝑥) − 𝑛𝛼 (𝑥) + 𝛼(𝑎) , Show that

𝑉𝛼 (𝑥) = 𝑝𝛼 (𝑥) + 𝑛𝛼 (𝑥) .

󵄨󵄨 𝑏 󵄨󵄨 𝑏 𝑏 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨∫ 𝑓 𝑑𝛼󵄨󵄨󵄨 ≤ ∫ |𝑓| 𝑑𝑝𝛼 + ∫ |𝑓| 𝑑𝑛𝛼 . 󵄨󵄨 𝑎 󵄨󵄨 𝑎 𝑎 󵄨 󵄨

Use this fact to again prove the estimate (4.44).

4.6 Exercises to Chapter 4 | 323

Exercise 4.45. Calculate the regularization 𝑓# for the function 𝑓 from Example 1.4. Exercise 4.46. Given 𝛼 ∈ 𝐵𝑉([𝑎, 𝑏]) and 𝑓 ∈ 𝐶([𝑎, 𝑏]), show that 󵄨󵄨 𝑏 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨∫ 𝛼 𝑑𝑓󵄨󵄨󵄨 ≤ ‖𝛼‖𝐵𝑉 ‖𝑓 − 𝑓(𝑏)‖∞ , 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑎 󵄨󵄨 where ‖ ⋅ ‖∞ denotes the norm (0.39). Exercise 4.47. Find an example of a function 𝛼 ∈ 𝐵𝑉𝑜 ([0, 1]) for which the inequality (4.64) is strict. Exercise 4.48. Discuss Example 0.24 from the viewpoint of Theorem 4.31. Exercise 4.49. For 1 < 𝑝 < ∞ and 𝑝󸀠 = 𝑝/(𝑝 − 1), prove directly (i.e. without making use of the abstract Theorem 0.23) that 𝑏

‖𝑓‖𝐿 𝑝

{ } = sup {∫ 𝑓(𝑥) 𝑑𝛼(𝑥) : 𝛼 ∈ 𝑅𝐵𝑉𝑜𝑝󸀠 ([𝑎, 𝑏]), Var𝑅𝑝󸀠 (𝛼; [𝑎, 𝑏]) ≤ 1} {𝑎 }

for all 𝑓 ∈ 𝐿 𝑝 ([𝑎, 𝑏]), and compare with Proposition 4.32. Exercise 4.50. For 0 < 𝜏 < 1, let 𝛼𝜏 : [0, 1] → ℝ be defined by 𝛼𝜏 (𝑥) = 𝑥𝜏 as in Example 2.78 and Example 3.35. Illustrate the abstract Theorems 4.33 and 4.35 for 𝛼 = 𝛼𝜏 .

5 Nonlinear composition operators Given a function ℎ : ℝ → ℝ, in this chapter, we study the nonlinear composition op­ erator 𝐶ℎ defined by 𝐶ℎ 𝑓 := ℎ ∘ 𝑓 for 𝑓 belonging to several spaces of functions of (generalized) bounded variation. It turns out that the typical condition, both neces­ sary and sufficient, for 𝐶ℎ to map such a space into itself is a local Lipschitz condi­ tion on ℎ. Afterwards, we briefly consider sufficient conditions on ℎ under which the corresponding operator 𝐶ℎ is bounded and/or continuous in norm; such conditions are often close to be also necessary. In view of applications, Lipschitz conditions for the operator 𝐶ℎ are of particular interest; in the last two sections, we will show that a global Lipschitz condition for 𝐶ℎ often leads to a strong degeneracy for ℎ, while a local Lipschitz condition for 𝐶ℎ is fulfilled for sufficiently large classes of nonlinear functions ℎ. Apart from our main object of study, the space 𝐵𝑉([𝑎, 𝑏]), we also con­ sider other function spaces which frequently arise in applications, such as spaces of Lipschitz, Hölder, or absolutely continuous functions.

5.1 The composition operator problem Let 𝑋 be some space of functions 𝑓 : [𝑎, 𝑏] → ℝ, and let ℎ : ℝ → ℝ be a fixed function. Under appropriate hypotheses on ℎ, we may then define a nonlinear operator 𝐶ℎ on 𝑋 by putting¹ (5.1) 𝐶ℎ 𝑓(𝑥) := ℎ(𝑓(𝑥)) (𝑎 ≤ 𝑥 ≤ 𝑏) . This operator is usually called the (autonomous) composition operator generated by ℎ. More generally, if ℎ : [𝑎, 𝑏] × ℝ → ℝ is a fixed function of two variables, one may also consider the (nonautonomous) superposition operator 𝑆ℎ 𝑓(𝑥) := ℎ(𝑥, 𝑓(𝑥))

(𝑎 ≤ 𝑥 ≤ 𝑏) .

(5.2)

It turns out that the behavior of the superposition operator (5.2) is far more compli­ cated than that of the composition operator (5.1). We postpone the study of the nonau­ tonomous operator (5.2) to the next Chapter 6; in this chapter, we confine ourselves to the autonomous operator (5.1). The most important problem related to the operator (5.1) reads as follows: – Given a function class 𝑋, find conditions on the function ℎ, possibly both necessary and sufficient, under which the operator 𝐶ℎ generated by ℎ maps the class 𝑋 into itself.

1 Sometimes, this operator is denoted by 𝐻, especially in the Russian literature, where it is often called Nemytskij operator. Recall that 𝐻 is the Russian letter for the Roman letter 𝑁; this fact allowed Hercule Poirot in Agatha Christie’s “Murder on the Orient Express” to discover the role of Princess Natalia Dragomiroff in the story.

5.1 The composition operator problem | 325

This problem is sometimes referred to as the composition operator problem (or COP, for short) in the literature. Here, we will use the notation 𝐶𝑂𝑃(𝑋) := {ℎ : 𝐶ℎ (𝑋) ⊆ 𝑋} = {ℎ : ℎ ∘ 𝑓 ∈ 𝑋 for all 𝑓 ∈ 𝑋} .

(5.3)

The problem of determining the set 𝐶𝑂𝑃(𝑋) for given 𝑋 is sometimes almost triv­ ial, and sometimes highly nontrivial. For example, it is easy to see that 𝐶𝑂𝑃(𝐶) = 𝐶, which means that the operator (5.1) maps the space 𝐶([𝑎, 𝑏]) into itself if and only if² the corresponding function ℎ is continuous on ℝ. This means, in particular, that the outer function ℎ may (and has to) be taken from the same class 𝐶 as the inner function 𝑓 to end up again in the class 𝐶. As we shall see later, for many function spaces 𝑋, the class 𝐶𝑂𝑃(𝑋) is essentially different from 𝑋 itself. In what follows, we will be interested in the problem of describing 𝐶𝑂𝑃(𝑋) for the various function classes 𝑋 which we introduced and studied in previous chapters. To begin with, let us make a trivial, though useful remark. Whenever the identity 𝑓(𝑥) = 𝑥 belongs to the class 𝑋 (and this is true for most function spaces), we automatically have 𝐶𝑂𝑃(𝑋) ⊆ 𝑋. The difficult part is of course to see whether or not this inclusion is strict, or we have equality ³ 𝐶𝑂𝑃(𝑋) = 𝑋. Before starting our discussion, we make a general remark. Denoting for the mo­ ment a space of functions 𝑓 : [𝑎, 𝑏] → ℝ by 𝑋([𝑎, 𝑏]), to emphasize the dependence on the domain of definition [𝑎, 𝑏], we introduce a special notion. Definition 5.1. We call a function space 𝑋 COP-invariant if, whenever the operator (5.1) maps the space 𝑋([𝑎, 𝑏]) into itself and is bounded in the norm of 𝑋([𝑎, 𝑏]) for some interval [𝑎, 𝑏], it also maps the space 𝑋([𝑐, 𝑑]) into itself and is bounded in the norm of 𝑋([𝑐, 𝑑]) for any other interval [𝑐, 𝑑]. ◼ The COP-invariance of a space 𝑋 justifies dropping the interval in the notation 𝐶𝑂𝑃(𝑋). The usual way to prove COP-invariance is by considering the strictly in­ creasing affine bijection ℓ(𝑡) :=

𝑏−𝑎 (𝑡 − 𝑐) + 𝑎 𝑑−𝑐

(𝑐 ≤ 𝑡 ≤ 𝑑)

(5.4)

(𝑎 ≤ 𝑠 ≤ 𝑏) ,

(5.5)

between [𝑐, 𝑑] and [𝑎, 𝑏] with inverse ℓ−1 (𝑠) =

𝑑−𝑐 (𝑠 − 𝑎) + 𝑐 𝑏−𝑎

2 The inclusion 𝐶 ⊆ 𝐶𝑂𝑃(𝐶) is an easy exercise in every first-year calculus course, while the inclusion 𝐶𝑂𝑃(𝐶) ⊆ 𝐶 uses the Tietze–Urysohn extension lemma, see Theorem 0.33. 3 To be precise, such an equality is not quite correct since 𝑋 consists of functions defined on [𝑎, 𝑏], while 𝐶𝑂𝑃(𝑋) consists of functions defined on the real line. The notation 𝐶𝑂𝑃(𝑋) ⊆ 𝑋 actually means 𝐶𝑂𝑃(𝑋([𝑎, 𝑏])) ⊆ 𝑋loc ((−∞, ∞)), i.e. ℎ belongs to the class 𝑋 over the real line if and only if ℎ|[𝑎,𝑏] ∈ 𝑋. This is the reason why we have to localize the function ℎ, as the formulation of Theorem 5.9 and similar theorems shows.

326 | 5 Nonlinear composition operators which we already considered in (0.80) and (0.81), and to show that it respects the struc­ ture of 𝑋. This leads to the following simple lemma which relates the COP-invariance of a function space 𝑋 with its shift-invariance introduced in Definition 0.45. Lemma 5.2. Every shift-invariant function space 𝑋 is COP-invariant. Proof. The proof is almost evident. The shift invariance of 𝑋 means that, for ℓ given by (5.4), the operator 𝐿𝑓 := 𝑓 ∘ ℓ is a linear isomorphism between 𝑋([𝑎, 𝑏]) and 𝑋([𝑐, 𝑑]) with inverse 𝐿−1 𝑔 = 𝑔∘ℓ−1 . So, if 𝐶ℎ : 𝑋([𝑎, 𝑏]) → 𝑋([𝑎, 𝑏]) is bounded, then 𝐿∘𝐶ℎ ∘𝐿−1 : 𝑋([𝑐, 𝑑]) → 𝑋([𝑐, 𝑑]) is bounded as well, and vice versa. This shows that 𝑋 is COPinvariant. We illustrate Definition 5.1 by means of four important examples. Example 5.3. We show that both the Wiener space 𝑋 = 𝑊𝐵𝑉𝑝 and the Riesz space 𝑋 = 𝑅𝐵𝑉𝑝 are COP-invariant for 𝑝 ≥ 1. To this end, by Lemma 5.2, it suffices to show that 𝑋 is shift-invariant. The function ℓ : [𝑐, 𝑑] → [𝑎, 𝑏] defined by (5.4) is an affine homeomorphism with inverse (5.5) which satisfies ℓ(𝑐) = 𝑎 and ℓ(𝑑) = 𝑏. Thus, ℓ : P([𝑐, 𝑑]) → P([𝑎, 𝑏]) with ℓ(𝑃) = ℓ({𝑡0 , 𝑡1 , . . . , 𝑡𝑚 }) = {ℓ(𝑡0 ), ℓ(𝑡1 ), . . . , ℓ(𝑡𝑚 )} defines a 1-1 correspondence between all partitions of [𝑐, 𝑑] and all partitions of [𝑎, 𝑏] since ℓ is strictly increasing. Consequently, for 𝑓 ∈ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]), we obtain 𝑚

𝑝 𝑊 Var𝑊 𝑝 (𝑓, ℓ(𝑃); [𝑎, 𝑏]) = ∑ |𝑓(ℓ(𝑡𝑗 )) − 𝑓(ℓ(𝑡𝑗−1 ))| = Var𝑝 (𝑓 ∘ ℓ, 𝑃; [𝑐, 𝑑]) . 𝑗=1

Passing to the supremum with respect to 𝑃 ∈ P([𝑐, 𝑑]) and ℓ(𝑃) ∈ P([𝑎, 𝑏]), we conclude that 𝑊 Var𝑊 (5.6) 𝑝 (𝑓; [𝑎, 𝑏]) = Var𝑝 (𝑓 ∘ ℓ; [𝑐, 𝑑]) , and so we have proved the shift-invariance of the space 𝑋 = 𝑊𝐵𝑉𝑝 . For the proof of the shift-invariance of the space 𝑋 = 𝑅𝐵𝑉𝑝 , the equality (5.6) has to be replaced with Var𝑅𝑝 (𝑓; [𝑎, 𝑏]) = (

𝑑 − 𝑐 𝑝−1 ) Var𝑅𝑝 (𝑓 ∘ ℓ; [𝑐, 𝑑]) . 𝑏−𝑎

The remaining part of the proof is the same.

(5.7) ♥

Observe that our reasoning in Example 5.3 gives even a more precise result than just COP-invariance: from (5.6) and ℓ(𝑐) = 𝑎, it follows that the map 𝑓 󳨃→ 𝑓∘ℓ is an isometry between 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) and 𝑊𝐵𝑉𝑝 ([𝑐, 𝑑]) for 𝑝 ≥ 1 (in particular, between 𝐵𝑉([𝑎, 𝑏]) and 𝐵𝑉([𝑐, 𝑑])). However, (5.7) shows that in case 𝑝 > 1, the map 𝑓 󳨃→ 𝑓 ∘ ℓ is not an isometry, but only an isomorphism between 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) and 𝑅𝐵𝑉𝑝 ([𝑐, 𝑑]).

5.1 The composition operator problem |

327

Example 5.4. Let us now show that the space 𝑋 = 𝐴𝐶 of absolutely continuous func­ tions is shift-invariant, and hence COP-invariant as well. Similarly, as before, we use the fact that the map ℓ from (5.4) defines, for all 𝑇 ∈ 𝛴([𝑐, 𝑑]), through the equality ℓ(𝑇) = ℓ({[𝑐1 , 𝑑1 ], [𝑐2 , 𝑑2 ], . . . , [𝑐𝑛 , 𝑑𝑛 ]}) = {[ℓ(𝑐1 ), ℓ(𝑑1 )], [ℓ(𝑐2 ), ℓ(𝑑2 )], . . . , [ℓ(𝑐𝑛 ), ℓ(𝑑𝑛 )]} a 1-1 correspondence between 𝛴([𝑐, 𝑑]) and 𝛴([𝑎, 𝑏]). So, for 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]), we have 𝑛

𝛤(𝑓; ℓ(𝑇)) = ∑ |𝑓(ℓ(𝑑𝑘 )) − 𝑓(ℓ(𝑐𝑘 ))| = 𝛤(𝑓 ∘ ℓ; 𝑇) ,

(5.8)

𝑘=1

where we use the notation (3.1). Given 𝜀 > 0, choose 𝜂 > 0 such that 𝛤(𝑓 ∘ ℓ; 𝑇) ≤ 𝜀 for all collections 𝑇 = {[𝑐1 , 𝑑1 ], [𝑐2 , 𝑑2 ], . . . , [𝑐𝑛 , 𝑑𝑛 ]} ∈ 𝛴([𝑐, 𝑑]) satisfying 𝛩(𝑇) ≤ 𝜂. Then 𝜂 and 𝑆 := ℓ(𝑇), we see that putting 𝛿 := 𝑏−𝑎 𝑑−𝑐 𝑛

𝛩(𝑆) = 𝛩(ℓ(𝑇)) = ∑ |ℓ(𝑑𝑘 ) − ℓ(𝑐𝑘 )| = 𝑘=1

𝑏−𝑎 𝛩(𝑇) . 𝑑−𝑐

Consequently, 𝛩(𝑆) ≤ 𝛿 implies 𝛩(𝑇) ≤ 𝜂, and hence 𝛤(𝑓; 𝑆) ≤ 𝜀, by (5.8). This shows that 𝑓 ∘ ℓ ∈ 𝐴𝐶([𝑐, 𝑑]), and so the space 𝐴𝐶 is shift-invariant. ♥ Example 5.4 admits a natural generalization which is given in Exercise 5.1. The follow­ ing example generalizes Example 5.3. Example 5.5. Let 𝜙 be a Young function which satisfies the condition ∞1 , see (2.16). We claim that the space 𝑋 = 𝑅𝐵𝑉𝜙 is COP-invariant. We could do this by imitating the reasoning of Example 5.3, but this would require to replace (5.7) by a suitable equality (or estimate) for Var𝑅𝜙 (𝑓; [𝑎, 𝑏]) and Var𝑅𝜙 (𝑓 ∘ ℓ; [𝑐, 𝑑]). Instead, we may prove the COPinvariance of 𝑋 = 𝑅𝐵𝑉𝜙 directly by using Medvedev’s theorem. Given ℎ : ℝ → ℝ, suppose that the operator (5.1) maps the space 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) into itself. We use the fact that the function ℓ : [𝑐, 𝑑] → [𝑎, 𝑏] defined by (5.4) is an affine > 0 and inverse (5.5). diffeomorphism with constant derivative ℓ󸀠 (𝑡) ≡ 𝑏−𝑎 𝑑−𝑐 Fix 𝑔 ∈ 𝑅𝐵𝑉𝜙 ([𝑐, 𝑑]). By Medvedev’s theorem (Theorem 3.36), we know that 𝑔 is absolutely continuous on [𝑐, 𝑑] and satisfies 𝑑

∫ 𝜙(𝛽|𝑔󸀠 (𝑡)|) 𝑑𝑡 < ∞ 𝑐

for some 𝛽 > 0. By what we have seen in Example 5.4, the function 𝑓 := 𝑔 ∘ ℓ−1 then 𝛽 and using the substitution 𝑠 = ℓ(𝑡), belongs to 𝐴𝐶([𝑎, 𝑏]). Moreover, putting 𝛼 := 𝑏−𝑎 𝑑−𝑐 we obtain 𝑏

𝑑

𝑏−𝑎 ∫ 𝜙(𝛼|𝑓 (𝑠)|) 𝑑𝑠 = ∫ 𝜙(𝛽|𝑔󸀠 (𝑡)|) 𝑑𝑡 < ∞ . 𝑑−𝑐 󸀠

𝑎

𝑐

Again, from Medvedev’s theorem, it follows that 𝑓 ∈ 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]), and so 𝐶ℎ 𝑓 = ℎ ∘ 𝑓 ∈ 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]), by assumption. So, 𝐶ℎ 𝑔 = ℎ ∘ 𝑓 ∘ ℓ ∈ 𝑅𝐵𝑉𝜙 ([𝑐, 𝑑]). ♥

328 | 5 Nonlinear composition operators Example 5.6. Finally, we show that the space 𝑋 = Λ𝐵𝑉 of functions of bounded Wa­ terman variation (Definition 2.15) is shift-invariant, and hence COP-invariant. In the same way as in Example 5.4, the affine homeomorphism ℓ maps 𝛴∞ ([𝑐, 𝑑]) into 𝛴∞ ([𝑎, 𝑏]) by means of the formula ℓ(𝑇∞ ) = ℓ({[𝑐1 , 𝑑1 ], [𝑐2 , 𝑑2 ], [𝑐3 , 𝑑3 ], . . .}) = {[ℓ(𝑐1 ), ℓ(𝑑1 )], [ℓ(𝑐2 ), ℓ(𝑑2 )], [ℓ(𝑐3 ), ℓ(𝑑3 )], . . .} . So, for 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]) and 𝑇∞ ∈ 𝛴∞ ([𝑐, 𝑑]), we have ∞

VarΛ (𝑓, ℓ(𝑇∞ ); [𝑎, 𝑏]) = ∑ 𝜆 𝑘 |𝑓(ℓ(𝑑𝑘 )) − 𝑓(ℓ(𝑐𝑘 ))| = VarΛ (𝑓 ∘ ℓ, 𝑇∞ ; [𝑐, 𝑑]) . 𝑘=1

Passing to the supremum with respect to 𝑇∞ ∈ 𝛴∞ ([𝑐, 𝑑]) and ℓ(𝑇∞ ) ∈ 𝛴∞ ([𝑎, 𝑏]), we conclude that VarΛ (𝑓; [𝑎, 𝑏]) = VarΛ (𝑓 ∘ ℓ; [𝑐, 𝑑]) , and so we have proved the shift-invariance of the space 𝑋 = 𝑊𝐵𝑉𝑝 .

♥

Again, our proof shows that the map 𝑓 󳨃→ 𝑓 ∘ ℓ is an isometry between Λ𝐵𝑉([𝑎, 𝑏]) and Λ𝐵𝑉([𝑐, 𝑑]). Before discussing the COP for the operator (5.1), in some spaces, we state a pre­ liminary result which is of independent interest [251]. Proposition 5.7. Suppose that the operator (5.1) maps the Waterman space Λ𝐵𝑉([𝑎, 𝑏]) into itself. Then the generating function ℎ : ℝ → ℝ is continuous. Proof. Let Λ = (𝜆 𝑛 )𝑛 be an arbitrary Waterman sequence, see Definition 2.15. Without loss of generality, we may assume that [𝑎, 𝑏] = [0, 1], ℎ(0) = 0, and ℎ is discontinuous at 0. Choose a positive decreasing sequence (𝑢𝑛)𝑛 which converges to 0 and satisfies ∞

ℎ(𝑢𝑛 ) ≥ 1, Let 𝑎𝑛 :=

∑ 𝑢𝑛 < 1 .

𝑛=1

2𝑛 + 1 , 2𝑛(𝑛 + 1)

𝑏𝑛 :=

1 , 𝑛

and consider the collection 𝑆∞ := {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([0, 1]). We define peak functions 𝑓 : [0, 1] → ℝ by 0 { { { 𝑓(𝑡) := {𝑢𝑛 { { {linear

for 𝑡 = 0 or 𝑡 = 𝑏𝑛 , 𝑛 = 1, 2, 3, . . . , for 𝑡 = 𝑎𝑛 , 𝑛 = 1, 2, 3, . . . , otherwise .

Note that then 𝑓 ∈ 𝐵𝑉([0, 1]), and hence 𝑓 ∈ Λ𝐵𝑉([0, 1]), by (2.33). On the other hand, |(ℎ ∘ 𝑓)(𝑏𝑛 ) − (ℎ ∘ 𝑓)(𝑎𝑛 )| = (ℎ ∘ 𝑓)(𝑎𝑛 ) = ℎ(𝑢𝑛 ) ≥ 1 , (5.9)

5.1 The composition operator problem

| 329

and therefore ∞

∞

∞

VarΛ (𝑓, 𝑆∞ ; [0, 1]) = ∑ 𝜆 𝑛|ℎ(𝑓(𝑏𝑛 )) − ℎ(𝑓(𝑎𝑛 ))| = ∑ 𝜆 𝑛 ℎ(𝑢𝑛 ) ≥ ∑ 𝜆 𝑛 = ∞ , 𝑛=1

𝑛=1

𝑛=1

by (5.9). This shows that 𝐶ℎ 𝑓 = ℎ ∘ 𝑓 does not belong to Λ𝐵𝑉([0, 1]), and so the proof is complete. We point out that Proposition 5.7 also holds for other spaces of functions of (general­ ized) bounded variation, see Exercise 5.8. Now, we discuss the COP for the operator (5.1) in the function spaces 𝐵𝑉, 𝑊𝐵𝑉𝑝 , 𝑅𝐵𝑉𝑝 , 𝐴𝐶, 𝐿𝑖𝑝, and 𝐿𝑖𝑝𝛼 . If 𝑋 denotes one of these spaces, we certainly have 𝐶𝑂𝑃(𝑋) ⊆ 𝑋loc (ℝ) since the identity 𝑓(𝑥) = 𝑥 belongs to 𝑋. It is a remarkable fact that except for 𝑋 = 𝐿𝑖𝑝, this inclusion is strict for all the other spaces 𝑋: Example 5.8. Let ℎ : ℝ → ℝ be the “seagull function” defined by {√|𝑢| for |𝑢| < 1 , ℎ(𝑢) := min {√|𝑢|, 1} = { 1 for |𝑢| ≥ 1 ; {

(5.10)

we will consider this function over and over for constructing counterexamples in what follows. Clearly, it suffices to consider ℎ on the interval [0, 1]. Of course, ℎ ∈ 𝐿𝑖𝑝1/2 ([0, 1]), and so ℎ ∈ 𝑊𝐵𝑉2 ([0, 1]), by (1.68). Moreover, our calculations in Exam­ ple 3.35 show that ℎ ∈ 𝑅𝐵𝑉𝑝 ([0, 1]) for 1 ≤ 𝑝 < 2 and Var𝑅𝑝 (ℎ; [0, 1]) =

1 2𝑝−1 (2 − 𝑝)

(1 ≤ 𝑝 < 2) .

So, from (2.93), we know that ℎ is also absolutely continuous.⁴ Of course, ℎ is not locally Lipschitz continuous on ℝ. We show now that the composition operator 𝐶ℎ generated by (5.10) does not map any of the spaces 𝑋 ∈ {𝐵𝑉, 𝑊𝐵𝑉𝑝 , 𝑅𝐵𝑉𝑝 , 𝐴𝐶, 𝐿𝑖𝑝, 𝐿𝑖𝑝𝛼 } into itself. The function 𝑓 : [0, 1] → ℝ defined by {𝑥2 sin2 𝑓(𝑥) := { 0 {

1 𝑥

for 0 < 𝑥 ≤ 1 , for 𝑥 = 0

(5.11)

belongs to 𝐿𝑖𝑝([0, 1]) since 𝑓 is differentiable with bounded derivative. On the other hand, the composed function 𝐶ℎ 𝑓 = ℎ ∘ 𝑓 does not even belong to 𝐵𝑉([0, 1]) since it is basically the function from Example 1.8. So, this example shows simultaneously that 𝐶ℎ (𝐿𝑖𝑝) ⊈ 𝐿𝑖𝑝, 𝐶ℎ (𝑅𝐵𝑉𝑝 ) ⊈ 𝑅𝐵𝑉𝑝 , 𝐶ℎ (𝐴𝐶) ⊈ 𝐴𝐶, and 𝐶ℎ (𝐵𝑉) ⊈ 𝐵𝑉.

4 The absolute continuity of ℎ may also be deduced from Theorem 3.9 since ℎ has the Luzin property and is continuous and monotone.

330 | 5 Nonlinear composition operators The simple function 𝑓(𝑥) := 𝑥𝛼 which belongs to 𝐿𝑖𝑝𝛼 ([0, 1]) is mapped by 𝐶ℎ into the function 𝐶ℎ 𝑓(𝑥) = 𝑥𝛼/2 which does not belong to 𝐿𝑖𝑝𝛼 ([0, 1]). To show that 𝐶ℎ (𝑊𝐵𝑉𝑝 ) ⊈ 𝑊𝐵𝑉𝑝 for general 𝑝, we use the special zigzag function 𝑍𝜃 defined in (0.93). According to Table 2.4, we have 𝑍𝜃 ∈ 𝑊𝐵𝑉𝑝 ([0, 1]) if and only if 𝑝𝜃 > 1. So, if we choose 1 < 𝑝𝜃 ≤ 2, then 𝑍𝜃 ∈ 𝑊𝐵𝑉𝑝 ([0, 1]), but 𝐶ℎ 𝑍𝜃 = ℎ ∘ 𝑍𝜃 ∈ ̸ 𝑊𝐵𝑉𝑝 ([0, 1]). Note that all of these examples also show that the requirement ℎ ∈ 𝐿𝑖𝑝𝛼 (ℝ) is not sufficient for ensuring that 𝐶ℎ maps any of the spaces 𝐵𝑉, 𝑊𝐵𝑉𝑝 , 𝑅𝐵𝑉𝑝 , 𝐴𝐶, 𝐿𝑖𝑝, or 𝐿𝑖𝑝𝛼 into itself. ♥ Interestingly, Example 5.8 also shows that the monotonicity requirement on 𝑓 is im­ portant in Proposition 2.79. In fact, the function 𝑓 in (5.11) belongs to 𝑅𝐵𝑉𝑝 for any 𝑝 ≥ 1, and the function ℎ in (5.10) belongs to 𝑅𝐵𝑉𝑞 for 𝑞 < 2. If Proposition 2.79 would apply in this case, we could choose in (2.154) any 𝑟 ≥ 1 satisfying 1−

1 1 1 < 1 ⋅ (1 − ) = , 𝑟 2 2

i.e. 𝑟 < 2 to guarantee that ℎ ∘ 𝑓 ∈ 𝑅𝐵𝑉𝑟 . However, we have seen above that ℎ ∘ 𝑓 does not even belong to 𝐵𝑉 = 𝑅𝐵𝑉1 . In terms of (5.3), we may summarize the contents of Example 5.8 in the set of in­ equalities 𝐶𝑂𝑃(𝐵𝑉) ≠ 𝐵𝑉loc (ℝ) ,

𝐶𝑂𝑃(𝑊𝐵𝑉𝑝 ) ≠ 𝑊𝐵𝑉𝑝,loc (ℝ) ,

𝐶𝑂𝑃(𝑅𝐵𝑉𝑝 ) ≠ 𝑅𝐵𝑉𝑝,loc (ℝ) , 𝐶𝑂𝑃(𝐿𝑖𝑝𝛼 ) ≠ 𝐿𝑖𝑝𝛼,loc (ℝ)

𝐶𝑂𝑃(𝐴𝐶) ≠ 𝐴𝐶loc (ℝ) , (0 < 𝛼 < 1) .

The reason for the bad behavior of the operator 𝐶ℎ in Example 5.8 will become clear in a moment: it is the lack of Lipschitz continuity of the seagull function ℎ. In fact, we shall show now in a series of theorems that, for 𝑝 ≥ 1 and 0 < 𝛼 ≤ 1, 𝐶𝑂𝑃(𝐵𝑉) = 𝐶𝑂𝑃(𝑊𝐵𝑉𝑝 ) = 𝐶𝑂𝑃(𝑅𝐵𝑉𝑝 )

(5.12)

= 𝐶𝑂𝑃(𝐴𝐶) = 𝐶𝑂𝑃(𝐿𝑖𝑝𝛼 ) = 𝐿𝑖𝑝loc (ℝ) . Throughout the following, the local Lipschitz condition |ℎ(𝑢) − ℎ(𝑣)| ≤ 𝑘(𝑟)|𝑢 − 𝑣| (𝑢, 𝑣 ∈ ℝ, |𝑢|, |𝑣| ≤ 𝑟)

(5.13)

will play a crucial role. Occasionally, we will also need the analogous condition |ℎ󸀠 (𝑢) − ℎ󸀠 (𝑣)| ≤ 𝑘1 (𝑟)|𝑢 − 𝑣| (𝑢, 𝑣 ∈ ℝ, |𝑢|, |𝑣| ≤ 𝑟)

(5.14)

for the derivative of ℎ (if it exists, of course). Together with the characteristics 𝑘(𝑟) and 𝑘1 (𝑟), the characteristics ̃ := sup |ℎ(𝑢)| (𝑟 > 0) (5.15) 𝑘(𝑟) |𝑢|≤𝑟

5.1 The composition operator problem

| 331

and 𝑘̃ 1 (𝑟) := sup |ℎ󸀠 (𝑢)| |𝑢|≤𝑟

(𝑟 > 0)

(5.16)

will also play a prominent role. The mean value theorem shows that these character­ istics are related by the simple estimates ̃ ≤ 𝑘(𝑟)𝑟 + |ℎ(0)|, 𝑘(𝑟)

𝑘̃ 1 (𝑟) ≤ 𝑘1 (𝑟)𝑟 + |ℎ󸀠 (0)| .

(5.17)

We start with the classical function space 𝐵𝑉([𝑎, 𝑏]). The COP for this space was completely solved by Josephy [155] who proved the following. Theorem 5.9. The operator (5.1) maps the space 𝐵𝑉([𝑎, 𝑏]) into itself if and only if the corresponding function ℎ is locally Lipschitz on ℝ, i.e. for each 𝑟 > 0, there exists 𝑘(𝑟) > 0 such that (5.13) holds. Proof. Although we will prove a more general result below (Theorem 5.10), we repro­ duce here Josephy’s original proof. Suppose first that ℎ satisfies (5.13) for each 𝑟 > 0. By the COP-invariance of the space 𝐵𝑉 (Example 5.3), we may assume without loss of generality that [𝑎, 𝑏] = [0, 1]. For 𝑓 ∈ 𝐵𝑉([0, 1]) and 𝑃 ∈ P([0, 1]) we have then Var(𝐶ℎ 𝑓, 𝑃; [0, 1]) ≤ 𝑘(‖𝑓‖∞ ) Var(𝑓, 𝑃; [0, 1]) ,

(5.18)

where ‖ ⋅ ‖∞ denotes the norm (0.39), and so 𝐶ℎ 𝑓 ∈ 𝐵𝑉([0, 1]) with ‖𝐶ℎ 𝑓‖BV ≤ |ℎ(𝑓(0))| + 𝑘(‖𝑓‖∞ )‖𝑓‖BV .

(5.19)

The nontrivial part is of course to show that the hypothesis 𝐶ℎ (𝐵𝑉) ⊆ 𝐵𝑉 implies the local Lipschitz condition (5.13). Suppose that ℎ ∈ ̸ 𝐿𝑖𝑝loc (ℝ); without loss of gener­ ality, we may assume that ℎ ∈ ̸ 𝐿𝑖𝑝([0, 1]). Then we may find sequences (𝑢𝑛 )𝑛 and (𝑣𝑛 )𝑛 in [0, 1] such that |ℎ(𝑢𝑛 ) − ℎ(𝑣𝑛 )| > (𝑛2 + 𝑛)𝛿𝑛 , (5.20) where 𝛿𝑛 := |𝑢𝑛 − 𝑣𝑛 |. Since the identity 𝑓(𝑡) = 𝑡 belongs to 𝐵𝑉([0, 1]), we know that ℎ ∈ 𝐵𝑉([0, 1]) and, in particular, ℎ is bounded on [0, 1], say |ℎ(𝑢)| ≤ 1/2 for 0 ≤ 𝑢 ≤ 1. So, from (5.20), we get (𝑛2 + 𝑛)𝛿𝑛 < |ℎ(𝑢𝑛 ) − ℎ(𝑣𝑛 )| ≤ 1 , and hence

1 (𝑛 = 1, 2, 3, . . .) . (5.21) 𝑛2 + 𝑛 Passing to appropriate subsequences if necessary, we may suppose that (𝑢𝑛 )𝑛 and (𝑣𝑛 )𝑛 converge to some point 𝑢∗ ∈ [0, 1] satisfying 𝛿𝑛
0. In particular, for any natural number 𝑛, we cannot have an estimate of the form |ℎ(𝑢) − ℎ(𝑣)| ≤ 𝑐|𝑢 − 𝑣| (𝑢, 𝑣 ∈ [−𝑟, 𝑟], |𝑢 − 𝑣| ≤ 𝑟/𝑛) since otherwise (5.13) would hold with 𝑘(𝑟) = 𝑛𝑐. Consequently, there exist 𝑢𝑘 < 𝑣𝑘 such that 𝛿𝑘 := 𝑣𝑘 − 𝑢𝑘
𝑘2 |𝑣𝑘 − 𝑢𝑘 | (𝑘 = 1, 2, . . .) .

(5.24)

Passing, if necessary, to a subsequence, we can assume without loss of generality that there exists 𝑢∞ ∈ [−𝑟, 𝑟] such that |𝑢𝑘 − 𝑢∞ | ≤ 1/2𝑘2 for all 𝑘, and so |𝑢𝑘 − 𝑢𝑘+1 | ≤ 1/𝑘2 . Take 𝑛𝑘 := ent (1/𝑘2 𝛿𝑘 ), i.e. 𝑛𝑘 is the unique natural number satisfying 1 1 ≤ 𝑛𝑘 < 2 + 1 (𝑘 = 1, 2, 3, . . .) . 𝑘2 𝛿𝑘 𝑘 𝛿𝑘

(5.25)

334 | 5 Nonlinear composition operators Then

1 2 + 𝛿𝑘 < 2 (𝑘 = 1, 2, 3, . . .) . 2 𝑘 𝑘 Consequently, the strictly increasing sequence (𝑡𝑘 )𝑘 defined recursively by 𝛿𝑘 𝑛𝑘 ≤

𝑡1 := 0,

𝑡𝑘+1 := 𝑡𝑘 + |𝑢𝑘 − 𝑢𝑘+1 | + 2𝑛𝑘 𝛿𝑘

is actually bounded by ∞

∞

5 < ∞. 𝑘2 𝑘=1

𝑇 := ∑ (|𝑢𝑘 − 𝑢𝑘+1 | + 2𝑛𝑘 𝛿𝑘 ) ≤ ∑ 𝑘=1

Now, we define a function 𝑓 : [0, 𝑇] → ℝ by 𝑢𝑘 { { { { { { {𝑣𝑘 { { 𝑓(𝑥) := {linear { { { { linear { { { { { 𝑢∞

if 𝑥 = 𝑡𝑘 + 2𝑚𝛿𝑘 for 𝑚 ∈ {0, . . . , 𝑛𝑘 } , if 𝑥 = 𝑡𝑘 + (2𝑚 − 1)𝛿𝑘 for 𝑚 ∈ {1, . . . , 𝑛𝑘 } , if 𝑡𝑘 + (𝑚 − 1)𝛿𝑘 ≤ 𝑥 ≤ 𝑡𝑘 + 𝑚𝛿𝑘 for 𝑚 ∈ {1, . . . , 2𝑛𝑘 } , if 𝑡𝑘 + 2𝑛𝑘 𝛿𝑘 ≤ 𝑥 ≤ 𝑡𝑘+1 , if 𝑥 = 𝑇 .

Note that |𝑢𝑘 − 𝑣𝑘 | = 𝛿𝑘 and |𝑢𝑘 − 𝑢𝑘+1 | ≤ |𝑡𝑘+1 − (𝑡𝑘 + 2𝑛𝑘 𝛿𝑘 )|, and so 𝑓 is actually Lipschitz continuous on [0, 𝑇) with Lipschitz constant 𝐿 ≤ 1. Consequently, 𝑓|[0,𝑇) has a unique continuous extension to a Lipschitz continuous function with Lipschitz constant 𝐿 on [0, 𝑇], and since 𝑢𝑘 → 𝑢∞ , actually 𝑓 itself is this extension. Assume now by contradiction that the operator (5.1) maps 𝐿𝑖𝑝([𝑎, 𝑏]) into 𝐵𝑉([𝑎, 𝑏]). Applying the result of Example 5.3 with [𝑐, 𝑑] = [0, 𝑢∞ ], this would imply that the func­ tion 𝐶ℎ 𝑓 = ℎ ∘ 𝑓 belongs to 𝐵𝑉([0, 𝑢∞ ]). On the other hand, by construction of 𝑓, (5.24) and (5.25), for any 𝑚 ∈ ℕ, we have 𝑚

𝑚

𝑘=1

𝑘=1

Var(ℎ ∘ 𝑓; [0, 𝑢∞ ]) ≥ ∑ 2𝑛𝑘 |ℎ(𝑣𝑘 ) − ℎ(𝑢𝑘 )| > ∑ 2𝑛𝑘 𝑘2 𝛿𝑘 ≥ 2𝑚 , which is a contradiction. This proves that statement (f) of Theorem 5.10 follows from (a), and so also any of the other statements (b), (c), (d), or (e). Conversely, we show now that (f) implies each of the statements (b), (c), (d) or (e), and so also statement (a). Assume that ℎ satisfies the local Lipschitz condition (5.13). Then 𝐶ℎ maps the space 𝐵𝑉([𝑎, 𝑏]) into itself, by Theorem 5.9, and so statement (d) is fulfilled. It remains to show that 𝐶ℎ maps each of the spaces 𝑋 = 𝐿𝑖𝑝([𝑎, 𝑏]) and 𝑋 = 𝐴𝐶([𝑎, 𝑏]) into itself. Indeed, for any function 𝑓 ∈ 𝑋, there is some 𝑟 with |𝑓(𝑡)| ≤ 𝑟 for all 𝑡 ∈ [𝑎, 𝑏], and thus 𝐶ℎ 𝑓 = ℎ ∘ 𝑓 satisfies |𝐶ℎ 𝑓(𝑠) − 𝐶ℎ 𝑓(𝑡)| = |ℎ(𝑓(𝑠)) − ℎ(𝑓(𝑡))| ≤ 𝑘(𝑟)|𝑓(𝑠) − 𝑓(𝑡)| and so also

𝑛

(𝑠, 𝑡 ∈ [𝑎, 𝑏]) ,

𝑛

∑ |𝐶ℎ 𝑓(𝑏𝑘 ) − 𝐶ℎ 𝑓(𝑎𝑘 )| ≤ 𝑘(𝑟) ∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|

𝑘=1

𝑘=1

5.1 The composition operator problem |

335

for every finite collection 𝑆 = {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([𝑎, 𝑏]). These estimates show that if 𝑓 is Lipschitz continuous or absolutely continuous, then 𝐶ℎ 𝑓 also has the re­ spective property. The proof is complete. Note that Theorem 5.10 covers not only the spaces occurring in the conditions (a)–(e), but any other “intermediate” space. We will prove this in the more general setting of the space 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) in Theorem 5.13 below. We also remark that Theorem 5.10 solves Exercise 3.5, see Exercise 5.7. One might wonder why the space 𝑊𝐵𝑉𝑝 ([0, 1]) is not covered by Theorem 5.10. In fact, one may show that (5.13) is also equivalent to the inclusion 𝐶ℎ (𝑊𝐵𝑉𝑝 ) ⊆ 𝑊𝐵𝑉𝑝 (see Theorem 5.12 below); this was recently proved in [15]. However, it is not true that (5.13) is equivalent to the inclusion 𝐶ℎ (𝐿𝑖𝑝) ⊆ 𝑊𝐵𝑉𝑝 which by (1.68) in case 𝑝 > 1 would be still weaker than (a): Example 5.11. Consider again the operator 𝐶ℎ generated by the seagull function (5.10). For any partition {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]), the estimate 𝑚 󵄨󵄨 󵄨󵄨2 (𝐶 𝑓, 𝑃; [0, 1]) = ∑ Var𝑊 󵄨󵄨󵄨ℎ(𝑓(𝑡𝑗 )) − ℎ(𝑓(𝑡𝑗−1 ))󵄨󵄨󵄨 ℎ 2 𝑗=1

𝑚

󵄨󵄨2 𝑚 󵄨 󵄨󵄨 󵄨 = ∑ 󵄨󵄨󵄨√|𝑓(𝑡𝑗 )| − √|𝑓(𝑡𝑗−1 )|󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨󵄨𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )󵄨󵄨󵄨󵄨 󵄨 󵄨 𝑗=1 𝑗=1 shows that the operator 𝐶ℎ maps 𝐵𝑉([0, 1]) into 𝑊𝐵𝑉2 ([0, 1]), and so also 𝐿𝑖𝑝([0, 1]) into 𝑊𝐵𝑉2 ([0, 1]). However, the function (5.10) certainly does not satisfy (5.13). ♥ Of course, the function (5.10) belongs to the Hölder space 𝐿𝑖𝑝1/2 ([0, 1]). So, it is not surprising that in order to include the family of spaces 𝑊𝐵𝑉 𝑝 ([0, 1]), we have to replace (5.13) by the local Hölder condition |ℎ(𝑢) − ℎ(𝑣)| ≤ 𝑘(𝑟)|𝑢 − 𝑣|𝛽

(𝑢 ∈ ℝ, |𝑢|, |𝑣| ≤ 𝑟)

(5.26)

for some fixed 𝛽 ∈ (0, 1]. We then obtain the following result which is parallel to The­ orem 5.10. Theorem 5.12. Any of the following four equivalent conditions on the operator (5.1) im­ plies condition (5.26) on the corresponding function ℎ, where 0 < 𝛽 ≤ 1: (a) The operator (5.1) maps 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) into 𝑊𝐵𝑉𝑝/𝛽 ([𝑎, 𝑏]). (b) The operator (5.1) maps 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) into 𝑊𝐵𝑉𝑞 ([𝑎, 𝑏]) for any 𝑞 ≥ 𝑝/𝛽. (c) The operator (5.1) maps 𝐵𝑉([𝑎, 𝑏]) into 𝑊𝐵𝑉1/𝛽 ([𝑎, 𝑏]). (d) The operator (5.1) maps 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) into 𝑊𝐵𝑉1/𝛼𝛽 ([𝑎, 𝑏]). Proof. The proof is similar to that of Theorem 5.10, and so we only sketch the idea. Since condition (a) implies (b), (b) implies (c), and (c) implies (d), by (1.68) and (1.72), we only have to prove that (d) implies (5.26). So, if we assume that (5.26) is false, we may find sequences (𝑢𝑘 )𝑘 and (𝑣𝑘 )𝑘 such that 1 |𝑢𝑘 − 𝑣𝑘 | ≤ 2 , |ℎ(𝑢𝑘 ) − ℎ(𝑣𝑘 )| > 𝑘2 |𝑢𝑘 − 𝑣𝑘 |𝛽 (𝑘 = 1, 2, . . .) , 𝑘

336 | 5 Nonlinear composition operators and then construct a function 𝑓 ∈ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) such that ℎ ∘ 𝑓 ∈ ̸ 𝑊𝐵𝑉1/𝛼𝛽 ([𝑎, 𝑏]) pre­ cisely in the same way as in the proof of Theorem 5.10. Of course, in case 𝛽 = 1, condition (5.26) reduces to condition (5.13), and we may recover from Theorem 5.12 some parts of Theorem 5.10. Thus, (a), (b) and (c) in The­ orem 5.12 then all reduce (for 𝑝 = 1) to (e) in Theorem 5.10, while (d) in Theorem 5.12 becomes (a) in Theorem 5.10 if, in addition, 𝛼 = 1. For 𝛽 = 1, we also get from (a) the equivalence of (5.13) and the inclusion 𝐶ℎ (𝑊𝐵𝑉𝑝 ) ⊆ 𝑊𝐵𝑉𝑝 . Note that, by Theo­ rem 5.12, the composition operator 𝐶ℎ generated by the seagull function (5.10) maps 𝑊𝐵𝑉𝑝 into 𝑊𝐵𝑉2𝑝 and 𝐿𝑖𝑝𝛼 into 𝑊𝐵𝑉2/𝛼 . Now, we give the complete solution of the COP for the spaces 𝑅𝐵𝑉𝜙 , Λ𝐵𝑉, and 𝑊𝐵𝑉𝜙 which is parallel to Theorem 5.9. Theorem 5.13 has been proved in [227], while Theorems 5.14 and 5.15 have been proved in [251]. Theorem 5.13. The operator (5.1) maps the space 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) into itself if and only if the corresponding function ℎ satisfies (5.13). Proof. In case 𝜙 ∈ ̸ ∞1 , we have 𝑅𝐵𝑉𝜙 = 𝐵𝑉, by Proposition 2.57, and the statement fol­ lows from Theorem 5.9. So, let us assume that 𝜙 ∈ ∞1 . By Example 5.5, we may suppose without loss of generality that [𝑎, 𝑏] = [0, 1]. First, let ℎ : ℝ → ℝ be locally Lipschitz on ℝ, and let 𝑓 ∈ 𝑅𝐵𝑉𝜙 ([0, 1]). Then 𝑓 is bounded on [0, 1] and Var𝑅𝜙 (𝜆𝑓; [0, 1]) < ∞ for some 𝜆 > 0 (Definition 2.55). Considering (5.13) for 𝑟 := ‖𝑓‖∞ , for any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]), we obtain the estimate 𝑚

∑𝜙( 𝑗=1 𝑚

≤ ∑𝜙( 𝑗=1

𝜆|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| |𝑡𝑗 − 𝑡𝑗−1 |

𝜆|ℎ(𝑓(𝑡𝑗 )) − ℎ(𝑓(𝑡𝑗−1 ))| 𝑘(‖𝑓‖∞ )|𝑡𝑗 − 𝑡𝑗−1 |

) |𝑡𝑗 − 𝑡𝑗−1 |

) |𝑡𝑗 − 𝑡𝑗−1 | ≤ Var𝑅𝜙 (𝜆𝑓; [0, 1]) < ∞ .

This shows that for 𝜇 := 𝜆/𝑘(‖𝑓‖∞ ), we get Var𝑅𝜙 (𝜇𝐶ℎ 𝑓; [0, 1]) < ∞, and hence 𝐶ℎ 𝑓 ∈ 𝑅𝐵𝑉𝜙 [0, 1] as claimed. The converse implication may be proved quite easily as a corollary of Theo­ rem 5.10. On the one hand, for 𝑓 ∈ 𝐿𝑖𝑝([𝑎, 𝑏]) with Lipschitz constant 𝐿 > 0 and any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), we have 𝑚

Var𝜙 (𝑓, 𝑃) = ∑ 𝜙 ( 𝑗=1

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| 𝑡𝑗 − 𝑡𝑗−1

) |𝑡𝑗 − 𝑡𝑗−1 |

𝑚

≤ ∑ 𝜙(𝐿)|𝑡𝑗 − 𝑡𝑗−1 | = 𝜙(𝐿)(𝑏 − 𝑎) , 𝑗=1

which shows that 𝑓 ∈ 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]), and hence 𝐿𝑖𝑝([𝑎, 𝑏]) ⊆ 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]). On the other hand, from Proposition 2.56, it follows that the inclusion 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) ⊆ 𝐵𝑉([𝑎, 𝑏]) is also true. So, the assertion follows from Theorem 5.10.

5.1 The composition operator problem

| 337

Now, we discuss the COP for the space Λ𝐵𝑉 of functions of bounded Waterman varia­ tion. Recall that a Waterman sequence is a decreasing sequence Λ = (𝜆 𝑛 )𝑛 converging to zero and satisfying (2.24). The corresponding function space Λ𝐵𝑉 has been intro­ duced in Definition 2.15. Theorem 5.14. The operator (5.1) maps the space Λ𝐵𝑉([𝑎, 𝑏]) into itself if and only if the corresponding function ℎ satisfies (5.13). Proof. The sufficiency of (5.13) is simple to prove. In fact, if ℎ satisfies (5.13) and 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]) with ‖𝑓‖Λ𝐵𝑉 ≤ 𝑟 is fixed, then for each 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]), we have ∞

VarΛ (ℎ ∘ 𝑓, 𝑆∞ ; [𝑎, 𝑏]) = ∑ 𝜆 𝑘 |ℎ(𝑓(𝑏𝑘 )) − ℎ(𝑓(𝑎𝑘 ))| 𝑘=1

(5.27)

∞

≤ 𝑘(𝑟) ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| = 𝑘(𝑟) VarΛ (𝑓, 𝑆∞ ; [𝑎, 𝑏]) , 𝑘=1

which shows that 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]) implies 𝐶ℎ 𝑓 = ℎ ∘ 𝑓 ∈ Λ𝐵𝑉([𝑎, 𝑏]). To prove the necessity of (5.13), suppose that ℎ does not satisfy a local Lipschitz condition. Without loss of generality, we may assume that there is a sequence (𝐽𝑛 )𝑛 of disjoint intervals 𝐽𝑛 = [𝑝𝑛 , 𝑞𝑛] with 𝑝𝑛 → 0 and 𝑞𝑛 → 0 as 𝑛 → ∞, as well as an unbounded real sequence (𝑐𝑛 )𝑛 , with 𝑐1 > 1, such that ∞

1 < ∞, 𝑛=1 𝑐𝑛 ∑

|ℎ(𝑞𝑛 ) − ℎ(𝑝𝑛 )| ≥ 𝑐𝑛 |𝑞𝑛 − 𝑝𝑛 | .

(5.28)

Since 𝑞𝑛 − 𝑝𝑛 → 0 and ℎ is continuous, by what we have observed before, we see that ℎ(𝑞𝑛 ) − ℎ(𝑝𝑛 ) → 0 as well, and so 𝑐𝑛 (𝑞𝑛 − 𝑝𝑛 ) → 0, by (5.28). Moreover, we may choose the intervals 𝐽𝑛 in such a way that the sequence (𝑐𝑛 (𝑞𝑛 − 𝑝𝑛 ))𝑛 is monotonically decreasing. Building on the sequences (𝐽𝑛 )𝑛 and (𝑐𝑛 )𝑛 , we now construct a function 𝑓 ∈ Λ𝐵𝑉([0, 1]) such that ℎ ∘ 𝑓 ∈ ̸ Λ𝐵𝑉([0, 1]). Fix a Waterman sequence Λ = (𝜆 𝑛)𝑛 , where without loss of generality, 𝜆 1 ≤ 1. The function 𝐿 : [0, ∞) → [0, ∞) defined by 0 { { { 𝐿(𝑥) := {𝜆[1, 𝑛] { { {linear

for 𝑥 = 0 , for 𝑥 = 𝑛 ∈ ℕ ,

(5.29)

otherwise ,

where we used the shortcut (2.42), is strictly increasing and piecewise linear with 𝐿(0) = 0 and 𝐿(𝑥) → ∞ as 𝑥 → ∞, and hence a bijection on [0, ∞). Denoting 𝑘𝑛 := ent 𝐿−1 (

1 + 1) 𝑐𝑛 (𝑞𝑛 − 𝑝𝑛 )

(𝑛 = 1, 2, 3, . . .) ,

where ent 𝜉 is the integer part of 𝜉, the monotonicity of 𝐿 implies that lim 𝑘𝑛 = ∞,

𝑛→∞

𝜆[1, 𝑘𝑛 ] = 𝐿(𝑘𝑛 ) >

1 , 𝑐𝑛 (𝑞𝑛 − 𝑝𝑛 )

(5.30)

338 | 5 Nonlinear composition operators and so |ℎ(𝑞𝑛 ) − ℎ(𝑝𝑛 )|𝜆[1, 𝑘𝑛 ] > 1, by (5.28). Choose 𝐶 > 1 such that 1 𝐶 < (𝑞𝑛 − 𝑝𝑛 )𝜆[1, 𝑘𝑛 ] < . 𝑐𝑛 𝑐𝑛

(5.31)

For this 𝐶, we then have ∞

∞

1 < ∞. 𝑐 𝑛=1 𝑛

∑ (𝑞𝑛 − 𝑝𝑛 )𝜆[1, 𝑘𝑛 ] < 𝐶 ∑

𝑛=1

(5.32)

For each 𝑛 ∈ ℕ, let {𝐼𝑛,1 , 𝐼𝑛,2 , . . . , 𝐼𝑛,𝑘𝑛 } be a finite collection of nonoverlapping in­ tervals contained in (2−𝑛 , 2−(𝑛−1) ) such that 𝐼𝑛,𝑚 is situated to the left of 𝐼𝑛,𝑚+1 for each 𝑚. Now, we define 𝑓 : [0, 1] → ℝ in the following way. First, we put 𝑓(0) = 𝑓(1) := 0. Next, for each 𝑛 ∈ ℕ, we define 𝑓 : [0, 1] → ℝ as an increasing linear map of 𝐼𝑛,𝑚 (𝑚 = 1, 2, . . . , 𝑘𝑛 ) onto 𝐽𝑛 . Finally, we define 𝑓 to be linear and continuous on the re­ maining components of [0, 1]. We claim that the function 𝑓 constructed in this way belongs to Λ𝐵𝑉([0, 1]). To prove this, we use Proposition 2.31. According to that proposition, it suffices to show that 𝐿 ∘ 𝐼𝑓 ∈ 𝐿 1 (ℝ), where 𝐿 is the function (5.29) and 𝐼𝑓 denotes the Banach indicatrix of 𝑓, see Definition 0.38. In fact, from the definition of 𝐿 in (5.29), it fol­ lows immediately that we may take 𝜇𝑓 (𝑦) := 𝜆[1, 𝐼𝑓 (𝑦)], where we use the notation of Proposition 2.31. Now, (5.32) shows that ∞

∞

∫ 𝜇𝑓 (𝑦) 𝑑𝑦 ≤ 𝑞1 𝜆[1, 2] + ∑ (𝑞𝑛 − 𝑝𝑛 )𝜆[1, 2𝑘𝑛 ] 𝑛=1

−∞

∞

≤ 𝑞1 𝜆[1, 2] + 2 ∑ (𝑞𝑛 − 𝑝𝑛 )𝜆[1, 𝑘𝑛 ] < ∞ , 𝑛=1

and hence 𝜇𝑓 ∈ 𝐿 1 (ℝ) and so 𝑓 ∈ Λ𝐵𝑉([0, 1]). It remains to show that 𝐶ℎ 𝑓 = ℎ ∘ 𝑓 ∈ ̸ Λ𝐵𝑉([0, 1]). To prove this, we write 𝐼𝑛,𝑚 = [𝑎𝑛,𝑚 , 𝑏𝑛,𝑚 ] and note that 𝑘𝑛

󵄨 󵄨 VarΛ (𝐶ℎ 𝑓; [2−𝑛 , 2−(𝑛−1) ]) ≥ ∑ 𝜆 𝑚 󵄨󵄨󵄨ℎ(𝑓(𝑏𝑛,𝑚 )) − ℎ(𝑓(𝑎𝑛,𝑚 ))󵄨󵄨󵄨 𝑚=1

(5.33)

= |ℎ(𝑞𝑛 ) − ℎ(𝑝𝑛 )|𝜆[1, 𝑘𝑛 ] > 1 , as we have seen above. If 𝐶ℎ 𝑓 were in Λ𝐵𝑉([0, 1]), then the right continuity of 𝐶ℎ 𝑓 at 0 would imply, by Exercise 2.14, that lim VarΛ (𝐶ℎ 𝑓; [2−𝑛 , 2−(𝑛−1) ]) = 0 ,

𝑛→∞

contradicting (5.33). Consequently, 𝐶ℎ 𝑓 does not belong to Λ𝐵𝑉([0, 1]), and the proof is complete. Now, we are ready to discuss the COP for the space 𝑊𝐵𝑉𝜙 of all functions of bounded Wiener–Young variation, at least for a restricted class of Young functions 𝜙. Recall that the 𝛿2 -condition was given in (2.4), condition ∞1 in (2.16), and condition 01 in (2.17).

5.1 The composition operator problem |

339

Theorem 5.15. Let 𝜙 be a Young function which satisfies a 𝛿2 -condition and condition ∞1 , but not condition 01 . Then the operator (5.1) maps the space 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) into itself if and only if the corresponding function ℎ satisfies (5.13). Proof. The proof is very similar to that of Theorem 5.14, and so we only sketch the general idea and the differences. Again, we may suppose without loss of generality that [𝑎, 𝑏] = [0, 1]. The fact that 𝑓 ∈ 𝑊𝐵𝑉𝜙 ([0, 1]) implies ℎ ∘ 𝑓 ∈ 𝑊𝐵𝑉𝜙 ([0, 1]) for ℎ satisfying (5.13) follows again from a simple calculation. To prove the necessity of (5.13), suppose that ℎ does not satisfy a local Lipschitz condition, and define the sequence (𝐽𝑛 )𝑛 of disjoint intervals 𝐽𝑛 = [𝑝𝑛 , 𝑞𝑛 ] and the unbounded real sequence (𝑐𝑛 )𝑛 satisfying (5.28) as in the proof of Theorem 5.14. However, the numbers 𝑘𝑛 from (5.30) will now be defined in a different way. Define a function 𝑃 : [0, ∞) → [0, ∞) by 0 { { { 𝑃(𝑥) := { 𝜙(𝑐 (𝑞1 −𝑝 )) { 𝑛 𝑛 𝑛 { {linear

for 𝑥 = 0 , for 𝑥 = 𝑛 ∈ ℕ , otherwise .

By construction, 𝑃 is then an increasing homeomorphism from [0, ∞) onto [0, ∞). Instead of (5.30), we now put 𝑘𝑛 := ent (𝑃(𝑛) + 1)

(𝑛 = 1, 2, 3, . . .) .

(5.34)

In particular, 𝑘𝑛 𝜙(𝑐𝑛 (𝑞𝑛 − 𝑝𝑛 )) > 1 for all 𝑛 ∈ ℕ. Now, let 𝑓 be defined as in the proof of Theorem 5.14, but with this new definition of the numbers 𝑘𝑛 . We claim that the function 𝑓 constructed in this way belongs to 𝑊𝐵𝑉𝜙 ([0, 1]), but 𝐶ℎ 𝑓 = ℎ ∘ 𝑓 does not. To prove the first assertion, let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]) be an arbitrary parti­ tion of [0, 1]; we have to find an upper bound for 𝑚

Var𝜙 (𝑓, 𝑃; [0, 1]) = ∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|)

(5.35)

𝑗=1

which does not depend on 𝑃. For the extreme indices 𝑗 = 1 and 𝑗 = 𝑚, we have 𝜙(|𝑓(𝑡1 ) − 𝑓(0)|) = 𝜙(|𝑓(𝑡1 )|) ≤ 𝜙(𝑞1 ) and 𝜙(|𝑓(1) − 𝑓(𝑡𝑚−1 )|) = 𝜙(|𝑓(𝑡𝑗−1 )|) ≤ 𝜙(𝑞1 ) , respectively. For the remaining indices 𝑗 ∈ {2, 3, . . . , 𝑚 − 1}, we distinguish two cases. Suppose first that there exists 𝑛 ∈ ℕ such that 𝑓(𝑡𝑗 ) ∈ 𝐽𝑛 and 𝑓(𝑡𝑗−1 ) ∈ 𝐽𝑛 . Writ­ ing again 𝐼𝑛,𝑚 = [𝑎𝑛,𝑚 , 𝑏𝑛,𝑚 ], we observe that 𝑎𝑛,1 ≤ 𝑡𝑗 , 𝑡𝑗−1 ≤ 𝑏𝑛,𝑘𝑛 . Clearly, 𝜙(|𝑓(𝑡𝑗 ) −

340 | 5 Nonlinear composition operators 𝑓(𝑡𝑗−1 )|) ≤ 𝜙(𝑞𝑛 − 𝑝𝑛 ). Note that increasing the number of partition points along inter­ vals of monotonicity does not increase the sum (5.35), and so the contribution of all such terms is bounded by 2𝑘𝑛 𝜙(𝑞𝑛 − 𝑝𝑛 ). Now, suppose that there is no 𝑛 ∈ ℕ such that 𝑓(𝑡𝑗 ) ∈ 𝐽𝑛 and 𝑓(𝑡𝑗−1 ) ∈ 𝐽𝑛 . Then there exists a smallest index 𝑘 satisfying 𝑓(𝑡𝑗 ) ≤ 𝑞𝑘 . Suppose that there are 𝑟 such intervals having this same smallest index 𝑘 satisfying the above.⁵ If [𝑡𝑗0 , 𝑡𝑗0 +1 ], [𝑡𝑗0 +1 , 𝑡𝑗0 +2 ], . . . , [𝑡𝑗0 +𝑟−1 , 𝑡𝑗0 +𝑟 ] are these intervals, we have, by the convexity of 𝜙, 𝑟

∑ 𝜙 (|𝑓(𝑡𝑗0 +𝑙 ) − 𝑓(𝑡𝑗0 +𝑙−1 )|) ≤ 𝜙 (|𝑓(𝑡𝑗0 +𝑟 ) − 𝑓(𝑡𝑗0 )|) ≤ 𝜙(𝑞𝑗 ) . 𝑙=1

By combining all of the obtained information, we see that (5.35) may be estimated from above by 𝑚

∞

∞

∑ 𝜙(|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|) ≤ 2𝜙(𝑞1 ) + 2 ∑ 𝑘𝑛 𝜙(𝑐𝑛 (𝑞𝑛 − 𝑝𝑛 )) + ∑ 𝜙(𝑞𝑛 ) 𝑛=1

𝑗=1

(5.36)

𝑛=1

which is obviously independent of the choice of the partition 𝑃. Observe that 𝑞𝑛 was chosen so that the last series in (5.36) converges. The other series in (5.36) may be estimated by ∞

∞

∑ 𝑘𝑛 𝜙(𝑐𝑛 (𝑞𝑛 − 𝑝𝑛 )) = ∑ ent (𝑃(𝑛) + 1)𝜙(𝑐𝑛 (𝑞𝑛 − 𝑝𝑛 ))

𝑛=1

𝑛=1 ∞

≤ ∑[ 𝑛=1 ∞

≤ ∑[ 𝑛=1 ∞

1 + 1] 𝜙(𝑞𝑛 − 𝑝𝑛 ) 𝜙(𝑐𝑛 (𝑞𝑛 − 𝑝𝑛 )) 𝜙(𝑞𝑛 − 𝑝𝑛 ) + 𝜙(𝑞𝑛 − 𝑝𝑛 )] 𝑐𝑛 𝑝ℎ𝑖(𝑞𝑛 − 𝑝𝑛 )

∞ ∞ ∞ 1 1 + ∑ 𝜙(𝑞𝑛 − 𝑝𝑛 ) ≤ ∑ + ∑ 𝜙(𝑞𝑛 ) < ∞ 𝑛=1 𝑐𝑛 𝑛=1 𝑛=1 𝑐𝑛 𝑛=1

=∑

where we have used the fact that 𝜙(𝑐𝑛 (𝑞𝑛 − 𝑝𝑛 )) ≥ 𝑐𝑛 𝜙(𝑞𝑛 − 𝑝𝑛 ) since 𝜙 is convex and 𝑐𝑛 ≥ 1. So, we have proved that 𝑓 ∈ 𝑊𝐵𝑉𝜙 ([0, 1]). It remains to show that 𝐶ℎ 𝑓 = ℎ ∘ 𝑓 ∈ ̸ 𝑊𝐵𝑉𝜙 ([0, 1]). Writing 𝐼𝑛,𝑚 = [𝑎𝑛,𝑚 , 𝑏𝑛,𝑚 ] and 𝐽𝑛 = [𝑝𝑛 , 𝑞𝑛 ] as before and taking 𝑘𝑛 as in (5.34), we get 𝑘𝑛

𝑘𝑛

∑ 𝜙 (|𝐶ℎ 𝑓(𝑏𝑛,𝑚 ) − 𝐶ℎ 𝑓(𝑎𝑛,𝑚 )|) = ∑ 𝜙 (|ℎ(𝑞𝑛 ) − ℎ(𝑝𝑛 ))|)

𝑗=1

𝑗=1

= 𝑘𝑛 𝜙 (|ℎ(𝑞𝑛 ) − ℎ(𝑝𝑛 )|) ≥ 𝑘𝑛 𝜙 (𝑐𝑛 (𝑞𝑛 − 𝑝𝑛 )) ≥

𝜙 (𝑐𝑛 (𝑞𝑛 − 𝑝𝑛 )) = 1. 𝜙 (𝑐𝑛 (𝑞𝑛 − 𝑝𝑛 ))

5 Observe that this situation can arise at most once for each index 𝑘.

5.1 The composition operator problem

| 341

Thus, for this collection of intervals 𝐼𝑛,𝑚 = [𝑎𝑛,𝑚 , 𝑏𝑛,𝑚 ], we have ∞ 𝑘𝑛

∞

∑ ∑ 𝜙 (|𝐶ℎ 𝑓(𝑏𝑛,𝑚 ) − 𝐶ℎ 𝑓(𝑎𝑛,𝑚 )|) ≥ ∑ 1 = ∞

𝑛=1 𝑗=1

𝑛=1

which shows that 𝐶ℎ = ℎ ∘ 𝑓 does not belong to 𝑊𝐵𝑉𝜙 ([0, 1]). The functions 𝜙(𝑡) = 𝑡𝑝 or 𝜙(𝑡) = (𝑡 + 1) log(𝑡 + 1) may serve as examples of Young functions which satisfy all the hypotheses of Theorem 5.15. We may summarize the contents of Theorems 5.13, 5.14 and 5.15 as equalities 𝐶𝑂𝑃(𝑅𝐵𝑉𝜙 ) = 𝐶𝑂𝑃(Λ𝐵𝑉) = 𝐶𝑂𝑃(𝑊𝐵𝑉𝜙 ) = 𝐿𝑖𝑝loc (ℝ)

(5.37)

which is completely analogous to (5.12). Again, our favorite function (5.10) may be used to show that the requirement ℎ ∈ 𝑋loc (ℝ) for 𝑋 ∈ {𝑅𝐵𝑉𝜙 , Λ𝐵𝑉, 𝑊𝐵𝑉𝜙 } does not imply 𝐶ℎ (𝑋) ⊆ 𝑋, i.e. ℎ ∈ 𝐶𝑂𝑃(𝑋). We have already proved this assertion in Example 5.8 for the space 𝑅𝐵𝑉𝑝 , and in Exam­ ple 5.11 for the space 𝑊𝐵𝑉𝑝 . So, it remains to show this for the Waterman space Λ𝐵𝑉; we state this as Example 5.16. Consider the Waterman space Λ 𝑞 𝐵𝑉([0, 1]) generated by the sequence 𝜆 𝑛 := 𝑛−𝑞 for 0 < 𝑞 ≤ 1, see Definition 2.29, and let ℎ : ℝ → ℝ be the seagull function (5.10). Then ℎ ∈ Λ 𝑞 𝐵𝑉([0, 1]) for any 𝑞 > 0 because ℎ ∈ 𝐵𝑉([0, 1]), see (2.33). In Section 2.2, we have shown that the special zigzag function 𝑍𝜃 defined in (0.93) belongs to Λ 𝑞 𝐵𝑉([0, 1]) if and only if 𝜃 + 𝑞 > 1; in this case, ∞

1 < ∞. 𝜃+𝑞 𝑛 𝑛=1

VarΛ 𝑞 (𝑍𝜃 ; [0, 1]) = ∑

So, if we choose 1 − 𝑞 < 𝜃 ≤ 2(1 − 𝑞), then⁶ 𝑍𝜃 ∈ Λ 𝑞 𝐵𝑉([0, 1]), but 𝐶ℎ 𝑍𝜃 = ℎ ∘ 𝑍𝜃 ∈ ̸ Λ 𝑞 𝐵𝑉([0, 1]). ♥ Our examples show again that the requirement ℎ ∈ 𝐿𝑖𝑝𝛼,𝑙𝑜𝑐 (ℝ) is not sufficient for the operator 𝐶ℎ to map any of the spaces 𝑅𝐵𝑉𝜙 , Λ𝐵𝑉 or 𝑊𝐵𝑉𝜙 into itself. One might ask why the proofs of Theorems 5.14 and 5.15 are so complicated and, in particular, why we could not use Theorem 5.10 to prove them, as we did for The­ orem 5.13. The point is that for applying Theorem 5.10, the spaces Λ𝐵𝑉([𝑎, 𝑏]) and 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) should be contained in 𝐵𝑉([𝑎, 𝑏]); however, as the relations (2.23) and (2.28) show, they contain 𝐵𝑉([𝑎, 𝑏]) as a (usually, strict) subspace. In the following table, where we use the shortcut [𝑎, 𝑏] = 𝐼 and [𝑐, 𝑑] = 𝐽, we make a comparison of different conditions of ℎ : 𝐽 → ℝ, under which the corresponding composition operator 𝐶ℎ maps all functions from some space 𝑋 = 𝑋(𝐼) whose range is contained in the domain 𝐽 of ℎ, into the same space 𝑋 = 𝑋(𝐼).

6 To be precise, we do not have ℎ ∘ 𝑍𝜃 = 𝑍𝜃/2 because this function is not piecewise linear; however, the statement ℎ ∘ 𝑍𝜃 ∉ Λ 𝑞 𝐵𝑉([0, 1]) follows from a direct calculation.

342 | 5 Nonlinear composition operators Table 5.1. Asymmetry in compositions of functions. 𝑓 ∈ 𝐵𝑉(𝐼), ℎ ∈ 𝐵𝑉(𝐽) 𝑓 ∈ 𝐵𝑉(𝐼), ℎ ∈ 𝐿𝑖𝑝𝛼 (𝐽)

󴁁󴁙󴀡 󴁁󴁙󴀡

ℎ ∘ 𝑓 ∈ 𝐵𝑉(𝐼) ℎ ∘ 𝑓 ∈ 𝐵𝑉(𝐼)

(Example 5.8) (Example 5.8)

𝑓 ∈ 𝐵𝑉(𝐼), ℎ ∈ 𝐿𝑖𝑝(𝐽) 𝑓 ∈ 𝑅𝐵𝑉𝑝 (𝐼), ℎ ∈ 𝑅𝐵𝑉𝑝 (𝐽)

⇔

ℎ ∘ 𝑓 ∈ 𝐵𝑉(𝐼)

(Theorem 5.9)

󴁁󴁙󴀡

ℎ ∘ 𝑓 ∈ 𝑅𝐵𝑉𝑝 (𝐼)

(Example 5.8)

𝑓 ∈ 𝑅𝐵𝑉𝑝 (𝐼), ℎ ∈ 𝐿𝑖𝑝𝛼 (𝐽)

󴁁󴁙󴀡

ℎ ∘ 𝑓 ∈ 𝑅𝐵𝑉𝑝 (𝐼)

(Example 5.8)

𝑓 ∈ 𝑅𝐵𝑉𝑝 (𝐼), ℎ ∈ 𝐿𝑖𝑝(𝐽)

⇔

ℎ ∘ 𝑓 ∈ 𝑅𝐵𝑉𝑝 (𝐼)

(Theorem 5.10)

𝑓 ∈ 𝑅𝐵𝑉𝜙 (𝐼), ℎ ∈ 𝑅𝐵𝑉𝜙 (𝐽)

󴁁󴁙󴀡

ℎ ∘ 𝑓 ∈ 𝑅𝐵𝑉𝜙 (𝐼)

(Example 5.8) (Example 5.8)

𝑓 ∈ 𝑅𝐵𝑉𝜙 (𝐼), ℎ ∈ 𝐿𝑖𝑝𝛼 (𝐽)

󴁁󴁙󴀡

ℎ ∘ 𝑓 ∈ 𝑅𝐵𝑉𝜙 (𝐼)

𝑓 ∈ 𝑅𝐵𝑉𝜙 (𝐼), ℎ ∈ 𝐿𝑖𝑝(𝐽)

⇔

ℎ ∘ 𝑓 ∈ 𝑅𝐵𝑉𝜙 (𝐼)

(Theorem 5.13)

𝑓 ∈ Λ𝐵𝑉(𝐼), ℎ ∈ Λ𝐵𝑉(𝐽) 𝑓 ∈ Λ𝐵𝑉(𝐼), ℎ ∈ 𝐿𝑖𝑝𝛼 (𝐽)

󴁁󴁙󴀡

ℎ ∘ 𝑓 ∈ Λ𝐵𝑉(𝐼)

(Example 5.16)

󴁁󴁙󴀡

ℎ ∘ 𝑓 ∈ Λ𝐵𝑉(𝐼)

(Example 5.16)

𝑓 ∈ Λ𝐵𝑉(𝐼), ℎ ∈ 𝐿𝑖𝑝(𝐽) 𝑓 ∈ 𝑊𝐵𝑉𝑝 (𝐼), ℎ ∈ 𝑊𝐵𝑉𝑝 (𝐽)

⇔

ℎ ∘ 𝑓 ∈ Λ𝐵𝑉(𝐼)

(Theorem 5.14)

󴁁󴁙󴀡

ℎ ∘ 𝑓 ∈ 𝑊𝐵𝑉𝑝 (𝐼)

(Example 5.8)

𝑓 ∈ 𝑊𝐵𝑉𝑝 (𝐼), ℎ ∈ 𝐿𝑖𝑝𝛼 (𝐽)

󴁁󴁙󴀡

ℎ ∘ 𝑓 ∈ 𝑊𝐵𝑉𝑝 (𝐼)

(Example 5.8)

𝑓 ∈ 𝑊𝐵𝑉𝑝 (𝐼), ℎ ∈ 𝐿𝑖𝑝(𝐽)

⇔

ℎ ∘ 𝑓 ∈ 𝑊𝐵𝑉𝑝 (𝐼)

(Theorem 5.12)

𝑓 ∈ 𝑊𝐵𝑉𝜙 (𝐼), ℎ ∈ 𝑊𝐵𝑉𝜙 (𝐽) 𝑓 ∈ 𝑊𝐵𝑉𝜙 (𝐼), ℎ ∈ 𝐿𝑖𝑝𝛼 (𝐽)

󴁁󴁙󴀡

ℎ ∘ 𝑓 ∈ 𝑊𝐵𝑉𝜙 (𝐼)

(Example 5.8)

󴁁󴁙󴀡

ℎ ∘ 𝑓 ∈ 𝑊𝐵𝑉𝜙 (𝐼)

(Example 5.8)

𝑓 ∈ 𝑊𝐵𝑉𝜙 (𝐼), ℎ ∈ 𝐿𝑖𝑝(𝐽) 𝑓 ∈ 𝐴𝐶(𝐼), ℎ ∈ 𝐴𝐶(𝐽)

⇔ 󴁁󴁙󴀡

ℎ ∘ 𝑓 ∈ 𝑊𝐵𝑉𝜙 (𝐼) ℎ ∘ 𝑓 ∈ 𝐴𝐶(𝐼)

(Theorem 5.15) (Example 5.8)

𝑓 ∈ 𝐴𝐶(𝐼), ℎ ∈ 𝐿𝑖𝑝𝛼 (𝐽) 𝑓 ∈ 𝐴𝐶(𝐼), ℎ ∈ 𝐿𝑖𝑝(𝐽)

󴁁󴁙󴀡 ⇔

ℎ ∘ 𝑓 ∈ 𝐴𝐶(𝐼) ℎ ∘ 𝑓 ∈ 𝐴𝐶(𝐼)

(Example 5.8) (Theorem 5.10)

𝑓 ∈ 𝐿𝑖𝑝𝛼 (𝐼), ℎ ∈ 𝐿𝑖𝑝𝛼 (𝐽) 𝑓 ∈ 𝐿𝑖𝑝𝛼 (𝐼), ℎ ∈ 𝐿𝑖𝑝(𝐽)

󴁁󴁙󴀡

ℎ ∘ 𝑓 ∈ 𝐿𝑖𝑝𝛼 (𝐼)

(Example 5.8)

⇔

ℎ ∘ 𝑓 ∈ 𝐿𝑖𝑝𝛼 (𝐼)

(Theorem 5.24)

This table exhibits a certain asymmetry in such conditions insofar as the require­ ment ℎ ∈ 𝑋 is never sufficient for guaranteeing that 𝐶ℎ (𝑋) ⊆ 𝑋. Roughly speaking, this table shows that local Lipschitz continuity of ℎ is the right condition, while local Hölder continuity does not suffice. The important point in Table 5.1 is of course that the crucial condition ℎ ∈ 𝐿𝑖𝑝([𝑐, 𝑑]) in every row where a theorem is cited is also necessary for the operator 𝐶ℎ to map the underlying space into itself. Thus, the equivalence arrow ⇔ in the third row, say, means that ℎ ∘ 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) for all functions 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) satis­ fying 𝑓([𝑎, 𝑏]) ⊆ [𝑐, 𝑑] if and only if ℎ ∈ 𝐿𝑖𝑝([𝑐, 𝑑]), and similarly for the other seven equivalence arrows. We remark that we did not prove the result mentioned in the last row of Table 5.1 for the Hölder space 𝐿𝑖𝑝 𝛼 ; for technical reasons, we postpone the proof to Theorem 5.24 below. However, we have already shown in Example 5.8 that 𝐿𝑖𝑝𝛼 is not stable under composition. Let us point out another important aspect of the crucial condition ℎ ∈ 𝐿𝑖𝑝loc (ℝ). We know that every function 𝑓 ∈ 𝐵𝑉 (and therefore also every function which belongs to 𝐿𝑖𝑝, 𝑅𝐵𝑉𝑝 , or 𝐴𝐶) is differentiable almost everywhere. So, the question arises as to

5.1 The composition operator problem |

343

whether or not the classical chain rule (ℎ ∘ 𝑓)󸀠 (𝑥) = ℎ󸀠 (𝑓(𝑥))𝑓󸀠 (𝑥)

(5.38)

holds under the hypotheses of Theorem 5.12, and in which sense it has to be inter­ preted. The following theorem answers this question for the space 𝐴𝐶. Theorem 5.17. Let 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) with 𝑓([𝑎, 𝑏]) ⊆ [𝑐, 𝑑], and let ℎ ∈ 𝐿𝑖𝑝([𝑐, 𝑑]). Then (5.38) holds for almost all 𝑥 ∈ [𝑎, 𝑏], where ℎ󸀠 (𝑓(𝑥))𝑓󸀠 (𝑥) is interpreted to be zero when­ ever 𝑓󸀠 (𝑥) = 0, even if ℎ is not differentiable at 𝑓(𝑥). Proof. Since 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) and 𝑔 := ℎ ∘ 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]), by Theorem 5.10, there exists a nullset 𝑁 ⊂ [𝑎, 𝑏] such that both 𝑓󸀠 (𝑥) and 𝑔󸀠 (𝑥) exist for 𝑥 ∈ [𝑎, 𝑏]\𝑁. Fix 𝑥0 ∈ [𝑎, 𝑏]\𝑁; we distinguish two cases. Suppose first that 𝑓󸀠 (𝑥0 ) = 0. Given 𝜀 > 0, we find then some ℎ > 0 such that |𝑓(𝑥) − 𝑓(𝑥0 )| ≤ 𝜀|𝑥 − 𝑥0 | for 𝑥 ∈ (𝑥0 − ℎ, 𝑥0 + ℎ) ∩ [𝑎, 𝑏]. It follows that |𝑔(𝑥) − 𝑔(𝑥0 )| ≤ 𝑙𝑖𝑝(ℎ)|𝑓(𝑥) − 𝑓(𝑥0 )| ≤ 𝑙𝑖𝑝(ℎ)𝜀|𝑥 − 𝑥0 | (|𝑥 − 𝑥0 | < ℎ) , and hence 𝑔󸀠 (𝑥0 ) = 0 = ℎ󸀠 (𝑓(𝑥0 ))𝑓󸀠 (𝑥0 ) as claimed. Now, suppose that 𝑓󸀠 (𝑥0 ) ≠ 0. For 𝑓(𝑥) ≠ 𝑓(𝑥0 ), we then have 𝑔(𝑥) − 𝑔(𝑥0 ) ℎ(𝑓(𝑥)) − ℎ(𝑓(𝑥0 )) 𝑓(𝑥) − 𝑓(𝑥0 ) = . 𝑥 − 𝑥0 𝑓(𝑥) − 𝑓(𝑥0 ) 𝑥 − 𝑥0

(5.39)

Since both 𝑓󸀠 (𝑥0 ) ≠ 0 and 𝑔󸀠 (𝑥0 ) exist, by assumption, we may pass to the limit for 𝑥 → 𝑥0 in (5.39) and obtain lim

𝑥→𝑥0

𝑔(𝑥) − 𝑔(𝑥0 ) 𝑔󸀠 (𝑥 ) = 󸀠 0 . 𝑓(𝑥) − 𝑓(𝑥0 ) 𝑓 (𝑥0 )

(5.40)

Denoting the right-hand side of (5.40) by 𝜌, for 𝜀 > 0, there exists some 𝛿 > 0 such that 𝜌−𝜀
0 such that any 𝑦 ∈ (𝑓(𝑥0 ) − 𝜂, 𝑓(𝑥0 ) + 𝜂) ∩ [𝑐, 𝑑] may be expressed as 𝑦 = 𝑓(𝑥) for some suitable 𝑥 ∈ (𝑥0 − 𝛿, 𝑥0 + 𝛿) ∩ [𝑎, 𝑏]. Putting this into (5.41) yields 𝜌−𝜀
0. In this case, the operator (5.1) is automatically bounded and continuous in the norm (0.11). The proof of Theorem 5.20 is a simple consequence of the Tietze–Urysohn extension theorem (Theorem 0.33), while the proof of Theorem 5.21 relies on a result of Kras­ nosel’skij, see [165–168] or [170]. Observe that the requirements on ℎ in Theorems 5.20 and 5.21 are quite different: in Theorem 5.20, we have to impose a simple analytic property (continuity of ℎ on ℝ), while in Theorem 5.21, we have to impose a growth condition (polynomial growth of ℎ of degree at most 𝑝/𝑞 for large values of 𝑢 ∈ ℝ). In the special case 𝑝 = 𝑞, the estimate (5.43) shows that ℎ has to be of sublinear growth for large values of 𝑢. Before passing to other spaces of continuous functions, let us briefly consider the COP for the most general space we have introduced in Chapter 0, namely, the space 𝑅([𝑎, 𝑏]) of regular functions. First, we show, by means of a counterexample, that the regularity of ℎ on the real axis is too weak to imply that the operator 𝐶ℎ maps the space 𝑅([𝑎, 𝑏]) into itself. Example 5.22. Let 𝐴 ⊂ [0, 1] be an uncountable Cantor set of positive measure (see, e.g. [76] or [118]), and define 𝑓 : [0, 1] → ℝ by 𝑓(𝑥) := dist(𝑥, 𝐴) = inf {|𝑥 − 𝑎| : 𝑎 ∈ 𝐴} .

7 Boundedness means that the operator maps bounded sets into bounded sets. We point out that in contrast to linear operators, a nonlinear operator like (5.1) or (5.2) may be bounded but discontinu­ ous, or continuous but unbounded. This is one of the reasons for the numerous difficulties which we encounter when dealing with these harmless looking operators.

346 | 5 Nonlinear composition operators Clearly, 𝑓 is (Lipschitz) continuous, and therefore regular. Moreover, the function ℎ := 𝜒{0} is certainly regular on ℝ. However, the composition 𝐶ℎ 𝑓 = ℎ ∘ 𝑓 = 𝜒𝐴 is not regular, by Proposition 0.35, since it is discontinuous on the uncountable set 𝐴. ♥ Example 5.22 shows even more than we claimed: continuity of ℎ on the whole real axis, except for one point, does not even imply the very weak condition 𝐶ℎ (𝐿𝑖𝑝) ⊆ 𝑅! It is somewhat surprising that the “right” condition for the inclusion 𝐶ℎ (𝑅) ⊆ 𝑅 is the same as for the inclusion 𝐶ℎ (𝐶) ⊆ 𝐶: Theorem 5.23. The operator (5.1) maps the space 𝑅([𝑎, 𝑏]) into itself if and only if the function ℎ is continuous on ℝ. In this case, the operator (5.1) is automatically bounded and continuous in the norm (0.39). Proof. Suppose first that ℎ : ℝ → ℝ is continuous, and let 𝑓 ∈ 𝑅([𝑎, 𝑏]). By Theo­ rem 0.36, we find a strictly increasing function 𝜏 : [𝑎, 𝑏] → [𝑎, 𝑏] and a continuous function 𝑔 : [𝑎, 𝑏] → ℝ such that 𝑓 = 𝑔 ∘ 𝜏. However, then the function 𝐶ℎ 𝑓 = (ℎ ∘ 𝑔) ∘ 𝜏 has an analogous representation, and again from Theorem 0.36, it follows that 𝐶ℎ 𝑓 is regular. Conversely, suppose now that ℎ : ℝ → ℝ is discontinuous at some point 𝑢0 ∈ ℝ, where without loss of generality⁸ ℎ(𝑢0 −) < ℎ(𝑢0 +), see (0.54). Choose 𝜀 > 0 and 𝛿 > 0 such that ℎ(𝑢) > ℎ(𝑢0 −) + 𝜀 for 𝑢0 < 𝑢 ≤ 𝑢0 + 𝛿, and define 𝑓 : [𝑎, 𝑏] → ℝ by 𝑓(𝑥) := ℎ(𝑢0 ) + min {dist(𝑥, 𝐴), 𝛿}

(𝑎 ≤ 𝑥 ≤ 𝑏) ,

where 𝐴 ⊂ [𝑎, 𝑏] is an uncountable Cantor set as in Example 5.22. Then 𝐶ℎ 𝑓 = ℎ ∘ 𝑓 is discontinuous at each point of 𝐴, and so it cannot belong to 𝑅([𝑎, 𝑏]). We still have to show that the operator 𝐶ℎ is bounded and continuous in the norm (0.39). Fix 𝑟 > 0 and 𝑓 ∈ 𝑅([𝑎, 𝑏]) with ‖𝑓‖∞ ≤ 𝑟. Then 𝑓(𝑥) ∈ [−𝑟, 𝑟] for 𝑎 ≤ 𝑥 ≤ 𝑏, and so ℎ(𝑓(𝑥)) ∈ [−𝑅, 𝑅] for 𝑎 ≤ 𝑥 ≤ 𝑏 and some 𝑅 > 0 since ℎ is continuous on the compact interval [−𝑟, 𝑟]. Similarly, the uniform continuity of ℎ on any compact interval implies that 𝐶ℎ is continuous at every point 𝑓 ∈ 𝑅([𝑎, 𝑏]) in the norm (0.39). For 0 < 𝛼 ≤ 1, we consider now the Banach space 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) of all Hölder continuous (in particular, Lipschitz continuous for 𝛼 = 1) functions on [𝑎, 𝑏], equipped with the usual norm (0.71) or the equivalent norm (0.77). Interestingly, in this space, we do not have a completely analogous result to Theorem 5.20 or Theorem 5.23: Theorem 5.24. The operator (5.1) maps the space 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) into itself if and only if the function ℎ satisfies (5.13). In this case, the operator (5.1) is automatically bounded in the norm (0.71).

8 Here, we use the fact that the assumption 𝐶ℎ (𝑅) ⊆ 𝑅 implies that ℎ is regular on the real line, and so both unilateral limits ℎ(𝑢0 −) and ℎ(𝑢0 +) exist.

5.2 Boundedness and continuity

| 347

Proof. First of all, we note that we cannot use Theorem 5.10 here in case 0 < 𝛼 < 1 since 𝐶ℎ (𝐿𝑖𝑝𝛼 ) ⊆ 𝐿𝑖𝑝𝛼 implies 𝐶ℎ (𝐿𝑖𝑝) ⊆ 𝐿𝑖𝑝𝛼 , but not 𝐶ℎ (𝐿𝑖𝑝) ⊆ 𝐵𝑉, as Example 1.23 shows. The sufficiency of (5.13) for 𝐶ℎ (𝐿𝑖𝑝𝛼 ) ⊆ 𝐿𝑖𝑝𝛼 is trivial; it is the necessity which requires a subtle construction. We already proved the necessity of (5.13) for 𝐶ℎ (𝐿𝑖𝑝) ⊆ 𝐿𝑖𝑝 in Theorem 5.10. So, let 0 < 𝛼 < 1, and assume that ℎ ∈ ̸ 𝐿𝑖𝑝loc (ℝ). As in the proof of Theorem 5.9, we can then find a constant 𝑟0 > 0 and two convergent sequences (𝑢𝑛 )𝑛 and (𝑣𝑛 )𝑛 in [−𝑟0 , 𝑟0 ] such that 𝑢𝑛 ≠ 𝑣𝑛 and |ℎ(𝑢𝑛 ) − ℎ(𝑣𝑛 )| > 𝑛|𝑢𝑛 − 𝑣𝑛 | (𝑛 = 1, 2, 3, . . .) .

(5.44)

Passing to subsequences, if necessary, we may assume that both 𝑢𝑛 → 𝑢∗ and 𝑣𝑛 → 𝑢∗ as 𝑛 → ∞. Consider the sequence of intervals 𝐼𝑛 := [𝑢∗ − 𝛾𝑛+1 , 𝑢∗ + 𝛾𝑛+1 ],

𝛾𝑛 :=

𝑏−𝑎 , 2𝑛0 +𝑛+2

where 𝑛0 ∈ ℕ is so large that 𝛾𝑛 < 1 for all 𝑛 ∈ ℕ. Then for every 𝑛, we find 𝑘𝑛 such that 𝑢𝑘𝑛 , 𝑣𝑘𝑛 ∈ 𝐼𝑛 which implies that 1/𝛼 = 𝛾𝑛1/𝛼 . 𝛿𝑛 := |𝑢𝑘𝑛 − 𝑣𝑘𝑛 |1/𝛼 < 21/𝛼 𝛾𝑛+1

This generates another sequence (𝛿𝑛 )𝑛 with 0 < 𝛿𝑛 < 𝛾𝑛 < 1. We define sequences (𝑠𝑛 )𝑛 and (𝑡𝑛 )𝑛 in [𝑎, 𝑏] by 𝑠1 := 𝑎 +

𝑏−𝑎 , 4

𝑠𝑛 := 𝑠1 + 𝛾1 + 𝛿1 + . . . + 𝛾𝑛−1 + 𝛿𝑛−1 ,

𝑡𝑛 := 𝑠𝑛 + 𝛿𝑛 .

Obviously, 𝛾𝑚 < 𝑠𝑛 − 𝑠𝑚 for 𝑛 > 𝑚. Since 0 < 𝜏 ≤ 1 implies 𝜏 ≤ 𝜏𝛼 and 𝜏 > 1 implies 𝜏 > 1, we find 𝛾𝑚 < (𝑠𝑛 −𝑠𝑚 )𝛼 for 𝑛 > 𝑚, and similarly 𝛾𝑚 < (𝑠𝑛 −𝑡𝑚 )𝛼 and 𝛾𝑚 < (𝑡𝑛 −𝑡𝑚 )𝛼 for 𝑛 > 𝑚. On the set 𝑀 := {𝑠1 , 𝑠2 , 𝑠3 , . . .} ∪ {𝑡1 , 𝑡2 , 𝑡3 , . . .}, we define a function 𝑓 by 𝛼

{𝑢𝑘 𝑓(𝑥) := { 𝑛 𝑣 { 𝑘𝑛

for 𝑥 = 𝑠𝑛 , for 𝑥 = 𝑡𝑛 .

A straightforward calculation shows that |𝑓(𝑠) − 𝑓(𝑡)| ≤ |𝑠 − 𝑡|𝛼

(𝑠, 𝑡 ∈ 𝑀) .

Applying now the McShane extension to 𝑓 (Theorem 0.42), we may extend 𝑓 to a function 𝑓 ∈ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) with Hölder constant 1. By assumption, the function 𝑔 := ℎ∘𝑓 belongs then also to 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]). So, there exists some 𝐿 > 0 satisfying |𝑔(𝑥) − 𝑔(𝑦)| ≤ 𝐿|𝑥 − 𝑦|𝛼

(𝑎 ≤ 𝑥, 𝑦 ≤ 𝑏) .

Putting, in particular, 𝑥 := 𝑠𝑛 and 𝑦 := 𝑡𝑛 in (5.45), we end up with |𝑔(𝑥) − 𝑔(𝑦)| = |ℎ(𝑢𝑘𝑛 ) − ℎ(𝑣𝑘𝑛 )| ≤ 𝐿|𝑠𝑛 − 𝑡𝑛 |𝛼 = 𝐿𝛿𝑛𝛼 = 𝐿|𝑢𝑘𝑛 − 𝑣𝑘𝑛 | .

(5.45)

348 | 5 Nonlinear composition operators Comparing this with (5.44) yields 𝐿 ≥ 𝑘𝑛 for all 𝑛 ∈ ℕ, a contradiction. So, we have proved that ℎ satisfies (5.13). The proof of the automatic boundedness of 𝐶ℎ is easy. In fact, ‖𝑓‖𝐿𝑖𝑝𝛼 ≤ 𝑟 implies ‖𝑓‖𝐶 ≤ 𝑟, and hence |𝐶ℎ 𝑓(𝑥) − 𝐶ℎ 𝑓(𝑦)| |𝑓(𝑥) − 𝑓(𝑦)| ≤ 𝑘(𝑟) ≤ 𝑘(𝑟)𝑙𝑖𝑝𝛼 (𝑓) ≤ 𝑘(𝑟)𝑟 , |𝑥 − 𝑦|𝛼 |𝑥 − 𝑦|𝛼 with 𝑘(𝑟) as in (5.13). So, 𝑙𝑖𝑝𝛼 (𝐶ℎ 𝑓) ≤ 𝑘(𝑟)𝑙𝑖𝑝𝛼 (𝑓) which shows that 𝐶ℎ is bounded in the norm (0.71). The reader may have noticed that in contrast to Theorems 5.20 and 5.21, we did not claim automatic continuity of the operator (5.1) in the space 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]). In fact, the operator 𝐶ℎ may map this space into itself without being continuous: Example 5.25. We give this counterexample for the case 𝛼 = 1, a similar example in case 0 < 𝛼 < 1 may be easily constructed. Let ℎ : ℝ → ℝ be defined by ℎ(𝑢) := min {|𝑢|, 1} .

(5.46)

Then the corresponding operator 𝐶ℎ maps 𝐿𝑖𝑝𝛼 ([0, 1]) into itself and is bounded, by Theorem 5.24. However, 𝐶ℎ is not continuous in the norm (0.71). To see this, con­ sider the functions 𝑓(𝑡) := 𝑡 and 𝑓𝑛 (𝑡) := 𝑡 + 1/𝑛 (𝑛 = 1, 2, 3, . . .). Clearly, ‖𝑓𝑛 − 𝑓‖𝐿𝑖𝑝𝛼 = |𝑓𝑛 (0) − 𝑓(0)| =

1 → 0 (𝑛 → ∞) . 𝑛

On the other hand, we have {𝑡 + 𝐶ℎ 𝑓𝑛 (𝑡) = { 1 { where 𝜏𝑛 := 1 − 1/𝑛, and hence 𝑙𝑖𝑝(𝐶ℎ 𝑓𝑛 − 𝐶ℎ 𝑓) ≥

1 𝑛

for 0 ≤ 𝑡 ≤ 𝜏𝑛 , for 𝜏𝑛 < 𝑡 ≤ 1 ,

|ℎ(𝑓𝑛 (𝜏𝑛 )) − ℎ(𝑓𝑛 (1)) − ℎ(𝑓(𝜏𝑛 )) − ℎ(𝑓(1)) 1 − 𝜏𝑛 = =1 1 − 𝜏𝑛 1 − 𝜏𝑛

which shows that ‖𝐶ℎ 𝑓𝑛 − 𝐶ℎ 𝑓‖𝐿𝑖𝑝 󴀀󴀂󴀠 0 as 𝑛 → ∞.

♥

One might ask which additional property of the function ℎ is “missing” in Theo­ rem 5.24 to also ensure the continuity of the operator 𝐶ℎ . Surprisingly, this question has a very natural answer: in [125], it was shown that 𝐶ℎ is continuous in 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) if and only if ℎ ∈ 𝐶1 (ℝ). We give a slightly more general result which states that this is even equivalent to the uniform continuity of 𝐶ℎ on bounded subsets.⁹

9 Recall that an operator 𝐴 : 𝑋 → 𝑌 between two normed spaces (𝑋, ‖ ⋅ ‖𝑋 ) and (𝑌, ‖ ⋅ ‖𝑌 ) is called uniformly continuous if for each 𝜀 > 0, one can find a 𝛿 > 0 such that ‖𝑓 − 𝑔‖𝑋 ≤ 𝛿 implies ‖𝐴𝑓 − 𝐴𝑔‖𝑌 ≤ 𝜀. If this holds for ‖𝑓‖𝑋 , ‖𝑔‖𝑋 ≤ 𝑟, and 𝛿 is allowed to depend on 𝑟, 𝐴 is called uniformly continuous on bounded sets. We will study these properties in detail in the next chapter for the superposition operator (5.2).

5.2 Boundedness and continuity

|

349

Theorem 5.26. The following three statements for 𝐶ℎ : 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) → 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) are equivalent: (a) The function ℎ belongs to 𝐶1 (ℝ). (b) The operator (5.1) is uniformly continuous on bounded subsets in the norm (0.71). (c) The operator (5.1) is continuous in the norm (0.71). Proof. To prove that (a) implies (b), suppose first that ℎ ∈ 𝐶1 (ℝ). Since this clearly implies (5.13), we know that 𝐶ℎ maps 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) into itself and is bounded in the norm (0.71). To show that 𝐶ℎ is uniformly continuous on bounded sets, let 𝑓, 𝑔 ∈ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) satisfy ‖𝑓‖𝐿𝑖𝑝𝛼 ≤ 𝑟 and ‖𝑔‖𝐿𝑖𝑝𝛼 ≤ 𝑟, and fix 𝜀 > 0. From ℎ ∈ 𝐶1 (ℝ), it follows that we find 𝛿 ∈ (0, 𝜀) such that ‖𝑓 − 𝑔‖𝐶 ≤ 𝛿 implies ‖𝐶ℎ 𝑓 − 𝐶ℎ 𝑔‖𝐶 ≤ 𝜀, and |𝑢 − 𝑣| ≤ 𝛿 implies |ℎ󸀠 (𝑢) − ℎ󸀠 (𝑣)| ≤ 𝜀 for 𝑢, 𝑣 ∈ [−𝑟, 𝑟]. So, we have, in particular, |𝐶ℎ 𝑓(𝑎) − 𝐶ℎ 𝑔(𝑎)| ≤ 𝜀, and it remains to estimate 𝑙𝑖𝑝𝛼 (ℎ ∘ 𝑓 − ℎ ∘ 𝑔). Using the Lagrange formula 1

ℎ(𝑢) − ℎ(𝑣) = (𝑢 − 𝑣) ∫ ℎ󸀠 [𝑣 + 𝜏(𝑢 − 𝑣)] 𝑑𝜏

(𝑢, 𝑣 ∈ ℝ),

0

for 𝑠 ≠ 𝑡, we obtain |𝐶ℎ 𝑓(𝑠) − 𝐶ℎ 𝑓(𝑡) − 𝐶ℎ 𝑔(𝑠) + 𝐶ℎ 𝑔(𝑡)| |𝑠 − 𝑡|𝛼 1

󵄨 󵄨 ≤ 𝑙𝑖𝑝𝛼 (𝑔) ∫ 󵄨󵄨󵄨󵄨ℎ󸀠 [𝑔(𝑠) + 𝜏(𝑔(𝑡) − 𝑔(𝑠))] − ℎ󸀠 [𝑓(𝑠) + 𝜏(𝑓(𝑡) − 𝑓(𝑠))]󵄨󵄨󵄨󵄨 𝑑𝜏 0

(5.47)

1

󵄨 󵄨 + 𝑙𝑖𝑝𝛼 (𝑔 − 𝑓) ∫ 󵄨󵄨󵄨󵄨ℎ󸀠 [𝑓(𝑠) + 𝜏(𝑓(𝑡) − 𝑓(𝑠))]󵄨󵄨󵄨󵄨 𝑑𝜏 . 0

Now, from |𝑓(𝑠) − 𝑔(𝑠)| ≤ 𝛿 and |𝑓(𝑡) − 𝑔(𝑡)| ≤ 𝛿, we get 󵄨 󵄨󵄨 󵄨󵄨𝑔(𝑠) + 𝜏(𝑔(𝑡) − 𝑔(𝑠)) − 𝑓(𝑠) + 𝜏(𝑓(𝑡) − 𝑓(𝑠))󵄨󵄨󵄨 ≤ (1 + 𝜏)|𝑓(𝑠) − 𝑔(𝑠)| + 𝜏|𝑓(𝑡) − 𝑔(𝑡)| ≤ 𝛿 , and so the first integrand in (5.47) may be estimated by 󵄨 󵄨󵄨 󸀠 󵄨󵄨ℎ [𝑔(𝑠) + 𝜏(𝑔(𝑡) − 𝑔(𝑠))] − ℎ󸀠 [𝑓(𝑠) + 𝜏(𝑓(𝑡) − 𝑓(𝑠))]󵄨󵄨󵄨 ≤ 𝜀 . 󵄨 󵄨 On the other hand, the second integrand remains bounded by 󵄨 󵄨󵄨 󸀠 󵄨󵄨ℎ [𝑓(𝑠) + 𝜏(𝑓(𝑡) − 𝑓(𝑠))]󵄨󵄨󵄨 ≤ 𝑘̃ 1 (𝑟) 󵄨 󵄨 ̃ with 𝑘 (𝑟) given by (5.16). Consequently, from 𝑙𝑖𝑝 (𝑓 − 𝑔) ≤ 𝛿, we conclude that 1

𝛼

|𝐶ℎ 𝑓(𝑠) − 𝐶ℎ 𝑓(𝑡) − 𝐶ℎ 𝑔(𝑠) + 𝐶ℎ 𝑔(𝑡)| ≤ 𝑙𝑖𝑝𝛼 (𝑔)𝜀 + 𝑘̃ 1 (𝑟)𝛿 ≤ (𝑙𝑖𝑝𝛼 (𝑔) + 𝑘̃ 1 (𝑟))𝜀 , |𝑠 − 𝑡|𝛼 and so we have proved (b).

350 | 5 Nonlinear composition operators The fact that (b) implies (c) is of course trivial; so it remains to prove that (c) implies (a). Suppose that the operator (5.1) maps 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) into itself and is both bounded and continuous. Since ℎ then satisfies (5.13), by Theorem 5.24, both limits 𝐿 − (𝑢) := lim inf ℎ→0

ℎ(𝑢 + ℎ) − ℎ(𝑢) , ℎ

𝐿 + (𝑢) := lim sup ℎ→0

ℎ(𝑢 + ℎ) − ℎ(𝑢) ℎ

exist and are finite for any 𝑢 ∈ ℝ. Moreover, the set 𝑁 := {𝑢 ∈ ℝ : 𝐿 − (𝑢) < 𝐿 + (𝑢)} is a nullset; we claim that 𝑁 = 0, and so ℎ󸀠 (𝑢) = 𝐿 − (𝑢) = 𝐿 + (𝑢) exists for all 𝑢 ∈ ℝ. Suppose that we can find 𝑢0 ∈ 𝑁 and choose two decreasing positive¹⁰ sequences (ℎ𝑛 )𝑛 and (𝑘𝑛 )𝑛 , both converging to zero, such that 𝐿 − (𝑢0 ) = lim

𝑛→∞

ℎ(𝑢0 + ℎ𝑛 ) − ℎ(𝑢0 ) , ℎ𝑛

𝐿 + (𝑢0 ) = lim

𝑛→∞

ℎ(𝑢0 + 𝑘𝑛 ) − ℎ(𝑢0 ) . 𝑘𝑛

Since 𝑁 is a nullset, we may find another sequence (𝛿𝑛 )𝑛 converging to zero such that ℎ󸀠 (𝑢0 + 𝛿𝑛 ) exists for all 𝑛 ∈ ℕ. Define functions 𝑓𝑛 : [𝑎, 𝑏] → ℝ by 𝑓𝑛 (𝑥) := (𝑥 − 𝑎)𝛼 + 𝑢0 + 𝛿𝑛

(𝑎 ≤ 𝑥 ≤ 𝑏, 𝑛 = 1, 2, 3, . . .) .

Clearly, all functions 𝑓𝑛 belong to 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]), and the sequence (𝑓𝑛 )𝑛 converges in the norm of 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) to the function 𝑓0 (𝑥) = (𝑥 − 𝑎)𝛼 + 𝑢0 which also belongs to 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]). Now, we use our continuity assumption on the operator 𝐶ℎ ; as a conse­ quence, we know that ‖𝐶ℎ 𝑓𝑛 − 𝐶ℎ 𝑓0 ‖𝐿𝑖𝑝𝛼 → 0 as 𝑛 → ∞. In particular, this implies that we may find 𝑛0 ∈ ℕ such that 𝑙𝑖𝑝𝛼 (ℎ ∘ 𝑓𝑛 − ℎ ∘ 𝑓0 ) ≤ 𝜀 :=

𝐿 + (𝑢0 ) − 𝐿 − (𝑢0 ) 4

for 𝑛 ≥ 𝑛0 . By definition of the functions 𝑓𝑛 and 𝑓0 , this means that |ℎ((𝑠 − 𝑎)𝛼 + 𝑢0 + 𝛿𝑛 ) − ℎ((𝑡 − 𝑎)𝛼 + 𝑢0 + 𝛿𝑛 ) − ℎ((𝑠 − 𝑎)𝛼 + 𝑢0 ) + ℎ((𝑡 − 𝑎)𝛼 + 𝑢0 )| ≤ 𝜀|𝑠 − 𝑡|𝛼 for all 𝑠, 𝑡 ∈ [𝑎, 𝑏] and 𝑛 ≥ 𝑛0 . Substituting now 𝑡 := 𝑎 and 𝑠 := 𝑎 + ℎ1/𝛼 𝑚 , we get 󵄨 󵄨󵄨 󵄨󵄨ℎ(𝑢0 + 𝛿𝑛 + ℎ𝑚 ) − ℎ(𝑢0 + 𝛿𝑛 ) − ℎ(𝑢0 + ℎ𝑚 ) − ℎ(𝑢0 )󵄨󵄨󵄨 ≤ 𝜀|ℎ𝑚 |

(5.48)

for 𝑛 ≥ 𝑛0 and 𝑚 sufficiently large. Dividing (5.48) by |ℎ𝑚 | and letting 𝑚 → ∞ yields |ℎ󸀠 (𝑢0 + 𝛿𝑛 ) − 𝐿 − (𝑢0 )| ≤ 𝜀 (𝑛 ≥ 𝑛0 ) .

(5.49)

10 The assumption ℎ𝑛 > 0 and 𝑘𝑛 > 0 may be made without loss of generality; otherwise, the following construction has to be modified accordingly.

5.2 Boundedness and continuity

| 351

The same argument with ℎ𝑚 replaced by 𝑘𝑚 shows that |ℎ󸀠 (𝑢0 + 𝛿𝑛 ) − 𝐿 + (𝑢0 )| ≤ 𝜀 (𝑛 ≥ 𝑛0 ) .

(5.50)

Finally, combining (5.49) and (5.50), we obtain 𝐿 + (𝑢0 ) − 𝐿 − (𝑢0 ) ≤ |𝐿 + (𝑢0 ) − ℎ󸀠 (𝑢0 + 𝛿𝑛 )| + |ℎ󸀠 (𝑢0 + 𝛿𝑛 ) − 𝐿 − (𝑢0 )| ≤ 2𝜀 =

1 [𝐿 (𝑢 ) − 𝐿 − (𝑢0 )] 2 + 0

for 𝑛 ≥ 𝑛0 , a contradiction. So, our assumption 𝑁 ≠ 0 was false, and ℎ󸀠 exists on the whole real line. Moreover, since (5.49) and (5.50) can be verified for any 𝑢0 ∈ ℝ, any 𝜀 > 0, and any sequence (𝛿𝑛 )𝑛 converging to zero, we may also conclude that ℎ󸀠 is continuous at every point, i.e. ℎ ∈ 𝐶1 (ℝ). Theorem 5.26 explains why the function (5.46) in Example 5.25 has to be chosen Lips­ chitz continuous, but not continuously differentiable at each point. Now, we turn to our main object of study, namely, the space 𝐵𝑉 and its various generalizations. The following result shows that also for these spaces, we get bound­ edness of 𝐶ℎ as a “fringe benefit.” Theorem 5.27. Let 1 ≤ 𝑝 < ∞, and let 𝑋 be any of the spaces 𝐴𝐶([𝑎, 𝑏]), 𝐵𝑉([𝑎, 𝑏]), 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]), or 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]). Suppose that the operator (5.1) maps the space 𝑋 into itself. Then this operator is automatically bounded in the norm of 𝑋. Proof. We know that the hypothesis 𝐶ℎ (𝑋) ⊆ 𝑋 implies that ℎ satisfies (5.13). For 𝑓 ∈ 𝑋 with ‖𝑓‖𝑋 ≤ 𝑟, we certainly have |𝑓(𝑎)| ≤ 𝑟, so (5.13) shows that |ℎ(𝑓(𝑎))| ≤ |ℎ(0)|+𝑘(𝑟)𝑟. Moreover, for any such 𝑓, we have |𝑓(𝑥)| ≤ 𝑟 for 𝑎 ≤ 𝑥 ≤ 𝑏, hence ‖𝑓‖∞ ≤ 𝑟. Given any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), by (5.13), we have, for 𝑝 ≥ 1, 𝑚

𝑚

𝑗=1

𝑗=1

∑ |ℎ(𝑓(𝑡𝑗 )) − ℎ(𝑓(𝑡𝑗−1 ))|𝑝 ≤ 𝑘(𝑟)𝑝 ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|𝑝 ,

𝑝 𝑊 which implies that Var𝑊 𝑝 (ℎ ∘ 𝑓) ≤ 𝑘(𝑟) Var𝑝 (𝑓); so 𝐶ℎ is bounded in the spaces 𝐵𝑉([𝑎, 𝑏]), 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]), and 𝐴𝐶([𝑎, 𝑏]). The analogous estimates 𝑚

∑ 𝑗=1

|ℎ(𝑓(𝑡𝑗 )) − ℎ(𝑓(𝑡𝑗−1 ))|𝑝 (𝑡𝑗 − 𝑡𝑗−1 )𝑝−1

𝑚

≤ 𝑘(𝑟)𝑝 ∑ 𝑗=1

|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|𝑝 (𝑡𝑗 − 𝑡𝑗−1 )𝑝−1

shows that 𝐶ℎ is bounded in the space 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) as well. We point out that Theorem 5.27 does not assert continuity of the operator 𝐶ℎ in any of the spaces covered by this theorem. In fact, continuity conditions which are both nec­ essary and sufficient seem to be unknown in these spaces. We give just one sufficient condition on the function ℎ which, in case of the spaces 𝐵𝑉 and 𝐴𝐶, guarantees the uniform continuity of 𝐶ℎ on bounded subsets.

352 | 5 Nonlinear composition operators Proposition 5.28. Suppose that ℎ ∈ 𝐿𝑖𝑝1loc (ℝ). Then the operator (5.1) is uniformly con­ tinuous on bounded subsets of 𝐵𝑉([𝑎, 𝑏]) and 𝐴𝐶([𝑎, 𝑏]). We do not give the proof here since we are going to prove a more general result in Theo­ rem 5.50 below. The following counterexample shows that the condition ℎ ∈ 𝐿𝑖𝑝loc (ℝ) is too weak to guarantee uniform continuity of 𝐶ℎ on bounded subsets of 𝐵𝑉 or 𝐴𝐶. Example 5.29. Let ℎ : ℝ → ℝ be defined by ℎ(𝑢) := |𝑢|. Since ℎ ∈ 𝐿𝑖𝑝loc (ℝ) (even ℎ ∈ 𝐿𝑖𝑝(ℝ)), the corresponding operator 𝐶ℎ maps each of the spaces 𝐵𝑉([0, 2𝜋]) and 𝐴𝐶([0, 2𝜋]) into itself and is bounded, by Theorem 5.10. Consider the sequences (𝑓𝑛 )𝑛 and (𝑔𝑛 )𝑛 defined by 𝑓𝑛 (𝑥) :=

cos 𝑛𝑥 , 𝑛

𝑔𝑛 (𝑥) := 𝑓𝑛 (𝑥) +

1 . 𝑛

Clearly, 𝑓𝑛 ∈ 𝐴𝐶([0, 2𝜋]) with Var(𝑓𝑛 ; [0, 2𝜋]) = 4, so both sequences (𝑓𝑛 )𝑛 and (𝑔𝑛 )𝑛 are bounded in the norm (1.16). Moreover, ‖𝑓𝑛 − 𝑔𝑛 ‖𝐵𝑉 → 0 as 𝑛 → ∞. For 𝑛 ∈ ℕ, consider the partition 𝑃𝑛 := {𝑡0 , 𝑡1 , . . . , 𝑡4𝑛 } ∈ P([0, 2𝜋]) defined by 𝑡0 := 0, 𝑡1 :=

𝑗𝜋 𝜋 , . . . , 𝑡𝑗 := , . . . , 𝑡4𝑛 := 2𝜋 . 2𝑛 2𝑛

An easy calculation then shows that 1 𝑓𝑛 (𝑡𝑗−1 ) = − , 𝑛

𝑓𝑛 (𝑡𝑗 ) = 𝑔𝑛 (𝑡𝑗−1 ) = 0,

𝑔𝑛 (𝑡𝑗 ) =

1 . 𝑛

Consequently, |𝐶ℎ 𝑓𝑛 (𝑡𝑗 ) − 𝐶ℎ 𝑔𝑛 (𝑡𝑗 ) − 𝐶ℎ 𝑓𝑛 (𝑡𝑗−1 ) + 𝐶ℎ 𝑔𝑛 (𝑡𝑗−1 )| 󵄨 󵄨 2 = 󵄨󵄨󵄨󵄨|𝑓𝑛 (𝑡𝑗 )| − |𝑔𝑛 (𝑡𝑗 )| − |𝑓𝑛 (𝑡𝑗−1 )| + |𝑔𝑛 (𝑡𝑗−1 )|󵄨󵄨󵄨󵄨 = , 𝑛 which implies that Var(𝐶ℎ 𝑓𝑛 − 𝐶ℎ 𝑔𝑛 ; [0, 2𝜋]) ≥ Var(𝐶ℎ 𝑓𝑛 − 𝐶ℎ 𝑔𝑛 , 𝑃𝑛 ; [0, 2𝜋]) 4𝑛

= ∑ |𝐶ℎ 𝑓𝑛 (𝑡𝑗 ) − 𝐶ℎ 𝑔𝑛 (𝑡𝑗 ) − 𝐶ℎ 𝑓𝑛 (𝑡𝑗−1 ) + 𝐶ℎ 𝑔𝑛 (𝑡𝑗−1 )| ≥ 8 , 𝑗=1

and so ‖𝐶ℎ 𝑓𝑛 − 𝐶ℎ 𝑔𝑛 ‖𝐵𝑉 󴀀󴀂󴀠 0 as 𝑛 → ∞.

♥

Let us summarize the boundedness and continuity behavior of the composition oper­ ator (5.1) discussed so far in a series of tables. Although these tables are not very excit­ ing in the case of the autonomous operator (5.1), we state them here in order to make a comparison with the nonautonomous operator (5.2) in Chapter 6 ( Tables 6.1–6.4). Of course, the situation is most satisfactory in Tables 5.2–5.4, where all state­ ments are equivalent. On the other hand, the Tables 5.6–5.8 do not contain conditions on ℎ, both necessary and sufficient, under which the operator 𝐶ℎ is continuous in the norm of the corresponding space. As far as we know, such continuity criteria are not known. Of course, one may try to find conditions which are just sufficient (Ex­

5.2 Boundedness and continuity | Table 5.2. The operator 𝐶ℎ in 𝐿 𝑝 ([𝑎, 𝑏]). 𝐶ℎ bounded in 𝐿 𝑝

⇔

𝐶ℎ (𝐿 𝑝 ) ⊆ 𝐿 𝑝 ⇕ |ℎ(𝑢)| ≤ 𝛼 + 𝛽|𝑢|

⇔

𝐶ℎ continuous in 𝐿 𝑝

Table 5.3. The operator 𝐶ℎ in 𝑅([𝑎, 𝑏]). 𝐶ℎ bounded in 𝑅

⇔

𝐶ℎ (𝑅) ⊆ 𝑅 ⇕ ℎ ∈ 𝐶(ℝ)

⇔

𝐶ℎ continuous in 𝑅

⇔

𝐶ℎ continuous in 𝐶

Table 5.4. The operator 𝐶ℎ in 𝐶([𝑎, 𝑏]). 𝐶ℎ bounded in 𝐶

⇔

𝐶ℎ (𝐶) ⊆ 𝐶 ⇕ ℎ ∈ 𝐶(ℝ)

Table 5.5. The operator 𝐶ℎ in 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) (0 < 𝛼 ≤ 1). 𝐶ℎ bounded in 𝐿𝑖𝑝𝛼

⇔

𝐶ℎ (𝐿𝑖𝑝𝛼 ) ⊆ 𝐿𝑖𝑝𝛼 ⇕ ℎ ∈ 𝐿𝑖𝑝loc (ℝ)

⇐ ⇐

𝐶ℎ continuous in 𝐿𝑖𝑝𝛼 ⇕ ℎ ∈ 𝐶1 (ℝ)

Table 5.6. The operator 𝐶ℎ in 𝐵𝑉([𝑎, 𝑏]). 𝐶ℎ bounded in 𝐵𝑉

⇔

𝐶ℎ (𝐵𝑉) ⊆ 𝐵𝑉 ⇕ ℎ ∈ 𝐿𝑖𝑝loc (ℝ)

⇐ ⇐

𝐶ℎ continuous in 𝐵𝑉 ⇑ ℎ ∈ 𝐿𝑖𝑝1loc (ℝ)

Table 5.7. The operator 𝐶ℎ in 𝐴𝐶([𝑎, 𝑏]). 𝐶ℎ bounded in 𝐴𝐶

⇔

𝐶ℎ (𝐴𝐶) ⊆ 𝐴𝐶 ⇕ ℎ ∈ 𝐿𝑖𝑝loc (ℝ)

⇐ ⇐

𝐶ℎ continuous in 𝐴𝐶 ⇑ ℎ ∈ 𝐿𝑖𝑝1loc (ℝ)

Table 5.8. The operator 𝐶ℎ in 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) (𝑝 > 1). 𝐶ℎ bounded in 𝑅𝐵𝑉𝑝

⇔

𝐶ℎ (𝑅𝐵𝑉𝑝 ) ⊆ 𝑅𝐵𝑉𝑝 ⇕ ℎ ∈ 𝐿𝑖𝑝loc (ℝ)

⇐

𝐶ℎ continuous in 𝑅𝐵𝑉𝑝

353

354 | 5 Nonlinear composition operators ercises 5.9 and 5.10). Conditions for the uniform continuity or uniform boundedness of 𝐶ℎ on bounded sets will be studied more systematically in Section 6.4 in the next chapter.

5.3 Spaces of differentiable functions Recall that given a space 𝑋 of functions 𝑓 : [𝑎, 𝑏] → ℝ with norm ‖ ⋅ ‖𝑋 , by 𝑋1 , we denote the space of all primitives of functions in 𝑋, i.e. 𝑋1 := {𝑓 : 𝑓󸀠 ∈ 𝑋}, equipped with the natural norm (5.51) ‖𝑓‖𝑋1 := |𝑓(𝑎)| + ‖𝑓󸀠 ‖𝑋 , see Definition 0.32. In this section, we shall study the composition operator (5.1) in such spaces for 𝑋 ∈ {𝐶, 𝐴𝐶, 𝐵𝑉, 𝑊𝐵𝑉𝑝 , 𝑅𝐵𝑉𝑝 , 𝐿𝑖𝑝}. By (5.51), the corresponding norms on these spaces are ‖𝑓‖𝐶1 = |𝑓(𝑎)| + max |𝑓󸀠 (𝑥)| ,

(5.52)

𝑎≤𝑥≤𝑏

𝑏 󸀠

‖𝑓‖𝐴𝐶1 = |𝑓(𝑎)| + |𝑓 (𝑎)| + ∫ |𝑓󸀠󸀠 (𝑡)| 𝑑𝑡 ,

(5.53)

𝑎 󸀠 1/𝑝 , ‖𝑓‖𝑊𝐵𝑉1𝑝 = |𝑓(𝑎)| + |𝑓󸀠 (𝑎)| + Var𝑊 𝑝 (𝑓 ; [𝑎, 𝑏]) 󸀠

‖𝑓‖𝑅𝐵𝑉1𝑝 = |𝑓(𝑎)| + |𝑓 (𝑎)| +

Var𝑅𝑝 (𝑓󸀠 ; [𝑎, 𝑏])1/𝑝

,

(5.54) (5.55)

and ‖𝑓‖𝐿𝑖𝑝1 = |𝑓(𝑎)| + |𝑓󸀠 (𝑎)| + 𝑙𝑖𝑝(𝑓󸀠 ; [𝑎, 𝑏]) ,

(5.56)

respectively. Taking, in particular, 𝑝 = 1 in (5.54) or (5.55), we get the space 𝐵𝑉1 with norm ‖𝑓‖𝐵𝑉1 = |𝑓(𝑎)| + |𝑓󸀠 (𝑎)| + Var(𝑓󸀠 ; [𝑎, 𝑏]) . (5.57) Interestingly, it turns out that the operator (5.1) does not behave in the space 𝑋1 in the same way as in the corresponding parent space 𝑋. We illustrate this by means of the composition operator problem described in Section 5.1: when analyzing the set (5.3) for given 𝑋, one may establish, loosely speaking, the following “golden rule” which applies quite frequently: – If not all functions in 𝑋 are differentiable, then 𝐶𝑂𝑃(𝑋) = 𝐿𝑖𝑝loc (ℝ). – If all functions in 𝑋 are differentiable, then 𝐶𝑂𝑃(𝑋) = 𝑋loc (ℝ). In other words, in the first case, the operator (5.1) maps 𝑋 into itself if and only if the corresponding function ℎ satisfies (5.13), and so the COP has an extrinsic and universal solution. We have proved this above for 𝑋 = 𝐵𝑉 in Theorem 5.9, for 𝑋 = 𝑅𝐵𝑉𝑝 , 𝑋 = 𝐿𝑖𝑝 and 𝑋 = 𝐴𝐶 in Theorem 5.10, for 𝑋 = 𝑊𝐵𝑉𝑝 in Theorem 5.12, for 𝑋 = 𝑅𝐵𝑉𝜙 in

5.3 Spaces of differentiable functions |

355

Theorem 5.13, for 𝑋 = Λ𝐵𝑉 in Theorem 5.14, for 𝑋 = 𝑊𝐵𝑉𝜙 in Theorem 5.15, and for 𝑋 = 𝐿𝑖𝑝𝛼 in Theorem 5.24. On the other hand, in the second case, the operator (5.1) maps 𝑋 into itself if and only if the corresponding function ℎ belongs (locally) to the same class, which is there­ fore an algebra with respect to composition, and so the COP has an intrinsic and indi­ vidual solution, as we will show now. As an example of the “golden rule,” we start with the simplest case 𝑋 = 𝐶, i.e. with the space 𝑋1 = 𝐶1 ([𝑎, 𝑏]) with norm (5.52). Theorem 5.30. The operator (5.1) maps the space 𝐶1 ([𝑎, 𝑏]) into itself if and only if the function ℎ is continuously differentiable on ℝ. In this case, the operator (5.1) is automat­ ically bounded and continuous in the norm (5.52). Proof. The proof is almost trivial. The inclusion 𝐶𝑂𝑃(𝐶1 ) ⊆ 𝐶1 (ℝ) follows from the fact that the identity 𝑓(𝑥) = 𝑥 is 𝐶1 , while the inclusion 𝐶𝑂𝑃(𝐶1 ) ⊇ 𝐶1 (ℝ) follows from the chain rule. To prove boundedness of 𝐶ℎ , we use the notation (5.15) and (5.16) for a function ℎ ∈ 𝐶1 (ℝ). From the chain rule, it follows then that ‖𝑓‖𝐶1 ≤ 𝑟 implies ̃ , |(ℎ ∘ 𝑓)(𝑎)| ≤ 𝑘(𝑟)

|(ℎ ∘ 𝑓)󸀠 (𝑡)| = |ℎ󸀠 (𝑓(𝑡))| |𝑓󸀠 (𝑡)| ≤ 𝑟𝑘̃ 1 (𝑟)

which shows that 𝐶ℎ is bounded. To prove that 𝐶ℎ is continuous in the norm (5.52), let (𝑓𝑛 )𝑛 be a sequence of 𝐶1 functions which converges in the norm (5.52) to some function 𝑓 ∈ 𝐶1 . Then (𝑓𝑛 )𝑛 is bounded, say ‖𝑓𝑛 ‖𝐶1 ≤ 𝑟. Putting 𝑔𝑛 := 𝐶ℎ 𝑓𝑛 − 𝐶ℎ 𝑓 = (ℎ ∘ 𝑓𝑛 ) − (ℎ ∘ 𝑓) ,

(5.58)

we get 𝑔𝑛󸀠 = (ℎ󸀠 ∘ 𝑓𝑛 )𝑓𝑛󸀠 − (ℎ󸀠 ∘ 𝑓)𝑓󸀠 , and we have to show that ‖𝑔𝑛󸀠 ‖𝐶 → 0 as 𝑛 → ∞. Let 𝜀 > 0. Since ℎ ∈ 𝐶1 (ℝ) and ‖𝑓𝑛 − 𝑓‖𝐶 → 0, by the mean value theorem, we may find 𝑛0 ∈ ℕ such that |ℎ󸀠 (𝑓𝑛 (𝑡)) − ℎ󸀠 (𝑓(𝑡))| ≤ 𝜀 for 𝑛 ≥ 𝑛0 and 𝑎 ≤ 𝑡 ≤ 𝑏. However, this implies that |𝑔𝑛󸀠 (𝑡)| ≤ |ℎ󸀠 (𝑓𝑛 (𝑡))| |𝑓𝑛󸀠 (𝑡) − 𝑓󸀠 (𝑡)| + |ℎ󸀠 (𝑓𝑛 (𝑡)) − ℎ󸀠 (𝑓(𝑡))| |𝑓󸀠 (𝑡)| ≤ 𝑘̃ (𝑟)‖𝑓󸀠 − 𝑓󸀠 ‖ + 𝜀‖𝑓󸀠 ‖ , 1

𝑛

𝐶

𝐶

which shows that ‖𝑔𝑛󸀠 ‖𝐶 → 0 as 𝑛 → ∞. The relation |𝑔𝑛 (𝑎)| → 0 is obvious. Now, we study the set 𝐶𝑂𝑃(𝑋1 ) for 𝑋 ∈ {𝐴𝐶, 𝐵𝑉, 𝑊𝐵𝑉𝑝 , 𝑅𝐵𝑉𝑝 , 𝐿𝑖𝑝}. As an immediate consequence of the inclusions (2.93), we get the inclusions 𝐿𝑖𝑝1 ([𝑎, 𝑏]) ⊆ 𝑅𝐵𝑉1𝑝 ([𝑎, 𝑏]) ⊆ 𝐴𝐶1 ([𝑎, 𝑏]) ⊆ 𝐶1 ([𝑎, 𝑏]) ∩ 𝐵𝑉1 ([𝑎, 𝑏]) for 𝑝 > 1. In the next example, we show that all inclusions in (5.59) are strict.

(5.59)

356 | 5 Nonlinear composition operators Example 5.31. For 𝜏 ≥ 1, consider the function¹¹ 𝑓𝜏 : [0, 1] → ℝ defined by 𝑓𝜏 (𝑥) := 𝑥𝜏 . A straightforward calculation shows that 𝑓𝜏 ∈ 𝐿𝑖𝑝1 ([0, 1]) ⇔ 𝑓𝜏−1 ∈ 𝐿𝑖𝑝([0, 1]) ⇔ 𝜏 ≥ 2 , 𝑓𝜏 ∈ 𝑅𝐵𝑉1𝑝 ([0, 1]) ⇔ 𝑓𝜏−1 ∈ 𝑅𝐵𝑉𝑝 ([0, 1]) ⇔ 𝜏 > 2 −

1 , 𝑝

and 𝑓𝜏 ∈ 𝐴𝐶1 ([0, 1]) ⇔ 𝑓𝜏−1 ∈ 𝐴𝐶([0, 1]) ⇔ 𝜏 > 1 . So, for 2− 𝑝1 < 𝜏 < 2, we have 𝑓𝜏 ∈ 𝑅𝐵𝑉1𝑝 ([0, 1])\𝐿𝑖𝑝1 ([0, 1]), while for 1 < 𝜏 ≤ 2− 𝑝1 ,

we have 𝑓𝜏 ∈ 𝐴𝐶1 ([0, 1]) \ 𝑅𝐵𝑉1𝑝 ([0, 1]). To show that the last inclusion in (5.59) is strict, we cannot use the function 𝑓𝜏 because the statements 𝑓𝜏 ∈ 𝐴𝐶1 , 𝑓𝜏 ∈ 𝐶1 , 𝑓𝜏 ∈ 𝐵𝑉1 , and 𝜏 > 1 are all equivalent. However, we may use the Cantor function 𝜑 : [0, 1] → ℝ from (3.6). Since 𝜑 is a non­ negative and continuous map of [0, 1] onto itself, the function 𝑓 : [0, 1] → ℝ defined by 𝑥

𝑓(𝑥) := ∫ 𝜑(𝑡) 𝑑𝑡

(0 ≤ 𝑥 ≤ 1)

(5.60)

0 1

is an increasing 𝐶 function. On the other hand, its derivative 𝑓󸀠 = 𝜑 does not map nullsets into nullsets, and thus fails to be absolutely continuous, by the Vital­ i–Banach–Zaretskij theorem (Theorem 3.9). Consequently, 𝑓 ∈ [𝐶1 ([0, 1])∩𝐵𝑉1 ([0, 1])]\ 𝐴𝐶1 ([0, 1]). ♥ Since we are going to deal with functions which have first derivatives everywhere and second derivatives almost everywhere, we will use the fact that under reasonable con­ ditions on 𝑓 and ℎ, we not have only the chain rule (5.38) for the first derivative, but also the chain rule (ℎ ∘ 𝑓)󸀠󸀠 (𝑥) = (ℎ󸀠󸀠 (𝑓(𝑥))𝑓󸀠2 (𝑥) + (ℎ(𝑓(𝑥))𝑓󸀠󸀠 (𝑥)

(5.61)

for the second derivative. Before studying the COP in spaces of differentiable functions different from 𝐶1 , we state a general lemma. Recall that 𝑃𝑛 ([𝑎, 𝑏]) denotes the linear space of all polynomials of degree ≤ 𝑛 on [𝑎, 𝑏]. Proposition 5.32. Suppose that a function space of type 𝑋1 = 𝑋1 ([𝑎, 𝑏]) contains the set 𝑃1 ([𝑎, 𝑏]) of all affine functions. Assume that ℎ : ℝ → ℝ generates a composition operator (5.1) which maps 𝑋1 into itself. Then ℎ ∈ 𝑋1loc (ℝ), i.e. the derivative ℎ󸀠 of ℎ belongs to the space 𝑋 = 𝑋([𝑐, 𝑑]) for any compact interval [𝑐, 𝑑] ⊂ ℝ.

11 For some properties of this function in case 𝜏 ≤ 1, see Examples 2.78 and 3.35.

5.3 Spaces of differentiable functions

| 357

Proof. Given [𝑐, 𝑑] ⊂ ℝ, consider the affine function ℓ−1 : [𝑎, 𝑏] → [𝑐, 𝑑] defined by (5.5). By assumption, ℓ−1 ∈ 𝑃1 ([𝑎, 𝑏]) ⊆ 𝑋1 ([𝑎, 𝑏]), and so 𝑔 := ℎ ∘ ℓ−1 ∈ 𝑋1 ([𝑎, 𝑏]) as well. Consequently, the function ℎ = 𝑔 ∘ ℓ belongs to 𝑋1 ([𝑐, 𝑑]) as claimed. Now, we discuss the COP for the space 𝐵𝑉1 ([𝑎, 𝑏]) of all primitives of 𝐵𝑉-functions equipped with the norm (5.57). Here, the situation is much more difficult than in the space 𝐶1 ([𝑎, 𝑏]). It is not hard to see that the condition ℎ ∈ 𝐿𝑖𝑝1loc (ℝ) is sufficient for 𝐶ℎ to map the space 𝐵𝑉1 ([𝑎, 𝑏]) into itself, while the condition ℎ ∈ 𝐵𝑉1loc (ℝ) is necessary, by Proposition 5.32. We begin with an example which shows that the condition ℎ ∈ 𝐿𝑖𝑝1loc (ℝ) is not necessary for the mapping condition 𝐶ℎ (𝐵𝑉1 ) ⊆ 𝐵𝑉1 , and hence too strong. Example 5.33. Let [𝑎, 𝑏] = [−1, 1] and ℎ(𝑢) := max {𝑢, 0}; clearly, ℎ󸀠 = 𝜒(0,∞) ∈ ̸ 𝐿𝑖𝑝loc (ℝ). On the other hand, for every 𝑓 ∈ 𝐵𝑉1 ([−1, 1]) ⊆ 𝐴𝐶([−1, 1]), we may apply (5.38) and get {𝑓󸀠 (𝑥) if 𝑓(𝑥) > 0 , (ℎ ∘ 𝑓)󸀠 (𝑥) = 𝑓󸀠 (𝑥)𝜒(0,∞) (𝑓(𝑥)) = { 0 if 𝑓(𝑥) ≤ 0 . { 󸀠 󸀠 Consequently, 𝑓 ∈ 𝐵𝑉([−1, 1]) implies (ℎ ∘ 𝑓) ∈ 𝐵𝑉([−1, 1]), and so 𝐶ℎ maps 1 𝐵𝑉 ([−1, 1]) into itself. ♥ We remark that the operator 𝐶ℎ in Example 5.33 is also bounded in the norm (5.57), and even continuous at zero since ‖𝐶ℎ 𝑓‖𝐵𝑉1 ≤ ‖𝑓‖𝐵𝑉1 . However, it is somewhat surprising that 𝐶ℎ is not continuous everywhere: Example 5.34. Let ℎ be defined as in Example 5.33, and let 1 (−1 ≤ 𝑥 ≤ 1) . 𝑓(𝑥) := 𝑥 , 𝑓𝑛 (𝑥) := 𝑥 + 𝑛 Clearly, 1 ‖𝑓𝑛 − 𝑓‖𝐵𝑉1 = |𝑓𝑛 (−1) − 𝑓(−1)| = → 0 (𝑛 → ∞) . 𝑛 The function 𝑔𝑛 := ℎ ∘ 𝑓𝑛 − ℎ ∘ 𝑓 and its derivative have the form 0 { { { 𝑔𝑛 (𝑥) = {𝑥 + { { 1 {𝑛 and 𝑔𝑛󸀠 (𝑥)

1 𝑛

for − 1 ≤ 𝑥 < − 𝑛1 , for −

1 𝑛

≤ 𝑥 < 0,

for 0 ≤ 𝑥 ≤ 1 ,

0 for − 1 ≤ 𝑥 < − 𝑛1 , { { { = {1 for − 𝑛1 ≤ 𝑥 < 0 , { { {0 for 0 ≤ 𝑥 ≤ 1 ,

respectively. Consequently, ‖𝑔𝑛 ‖𝐵𝑉1 ≥ Var(𝑔𝑛󸀠 ; [−1, 1]) = 2 , which shows that ℎ ∘ 𝑓𝑛 󴀀󴀂󴀠 ℎ ∘ 𝑓 in the norm (5.57) as 𝑛 → ∞.

♥

358 | 5 Nonlinear composition operators The following remarkable Theorems 5.35 and 5.37 which are due to Burenkov [74, 75] give a complete solution of the COP in the spaces 𝐵𝑉1 and 𝐴𝐶1 . Theorem 5.35. The operator (5.1) maps the space 𝐵𝑉1 ([𝑎, 𝑏]) into itself if and only if ℎ ∈ 𝐵𝑉1loc (ℝ). Moreover, in this case, the operator (5.1) is automatically bounded in the norm (5.57). Proof. The “only if” part is a consequence of Proposition 5.32, and so we must only prove the “if” part. Let 𝑓 ∈ 𝐵𝑉1 ([𝑎, 𝑏]), ℎ ∈ 𝐵𝑉1 ([𝑐, 𝑑]), where 𝑓([𝑎, 𝑏]) ⊆ [𝑐, 𝑑], and 𝑔 := ℎ ∘ 𝑓. First, we note that the function 𝑓󸀠 is then continuous: since 𝑓󸀠 has a primi­ tive, the Darboux intermediate value theorem implies that 𝑓󸀠 cannot have removable discontinuities or discontinuities of first kind (jumps), and since 𝑓󸀠 is of bounded vari­ ation, 𝑓󸀠 cannot have discontinuities of second kind either.¹² So, we may apply Exer­ cise 1.53 for 𝜙 := ℎ󸀠 ∘ 𝑓 and 𝜓 := 𝑓󸀠 and obtain ∞

Var(𝑔󸀠 ; [𝑎, 𝑏]) ≤ ‖ℎ󸀠 ∘ 𝑓‖∞ Var(𝑓󸀠 ; [𝑎, 𝑏]) + ∑ Var(𝑔󸀠 ; [𝑎𝑘 , 𝑏𝑘 ]) ,

(5.62)

𝑘=1

where (𝑎𝑘 , 𝑏𝑘 ) denote the connected components of (𝑎, 𝑏) \ (𝑓󸀠 )−1 (0), i.e. the open inter­ vals between two subsequent zeros of 𝑓󸀠 . Since ℎ󸀠 ∘ 𝑓 is bounded on [𝑎, 𝑏], being the composition of two bounded functions, and Var(𝑓󸀠 ; [𝑎, 𝑏]) is finite, by assumption, the only term we have to estimate is the series (or sum) in (5.62). Given 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎𝑘 , 𝑏𝑘 ]), we get 𝑚

Var(𝑔󸀠 , 𝑃; [𝑎𝑘 , 𝑏𝑘 ]) = ∑ |𝑔󸀠 (𝑡𝑗 ) − 𝑔󸀠 (𝑡𝑗−1 )| 𝑗=1

𝑚

𝑚 󵄨 󵄨 󵄨 󵄨 ≤ ∑ 󵄨󵄨󵄨󵄨ℎ󸀠 (𝑓(𝑡𝑗 )) − ℎ󸀠 (𝑓(𝑡𝑗−1 ))󵄨󵄨󵄨󵄨 |𝑓󸀠 (𝑡𝑗 )| + ∑ 󵄨󵄨󵄨󵄨ℎ󸀠 (𝑓(𝑡𝑗−1 ))󵄨󵄨󵄨󵄨 |𝑓󸀠 (𝑡𝑗 ) − 𝑓󸀠 (𝑡𝑗−1 )| 𝑗=1

𝑗=1

󸀠

󸀠

≤ sup |𝑓 (𝑡𝑗 )| Var(ℎ ∘ 𝑓, 𝑃; [𝑎𝑘 , 𝑏𝑘 ]) + sup |ℎ󸀠 (𝑢)| Var(𝑓󸀠 , 𝑃; [𝑎𝑘 , 𝑏𝑘 ]) 󸀠

󸀠

𝑗=0,...,𝑚

𝑐≤𝑢≤𝑑

≤ sup |𝑓 (𝑡𝑗 )| Var(ℎ ; [𝑐, 𝑑]) + sup |ℎ (𝑢)| Var(𝑓󸀠 , 𝑃; [𝑎𝑘 , 𝑏𝑘 ]) , 𝑗=0,...,𝑚

󸀠

𝑐≤𝑢≤𝑑

where in the last inequality, we have used the fact that 𝑓󸀠 (𝑥) ≠ 0 on (𝑎𝑘 , 𝑏𝑘 ), and hence 𝑓 is monotone on [𝑎𝑘 , 𝑏𝑘 ]. We distinguish two cases. If 𝑓󸀠 (𝑥) ≠ 0 on [𝑎, 𝑏], then the series in (5.62) consists of just one term, and the above estimate simplifies to Var(𝑔󸀠 ; [𝑎, 𝑏]) ≤ ‖𝑓󸀠 ‖𝐶 Var(ℎ󸀠 ; [𝑐, 𝑑]) + sup |ℎ󸀠 (𝑢)| Var(𝑓󸀠 ; [𝑎, 𝑏]) ,

(5.63)

𝑐≤𝑢≤𝑑

12 This reasoning implies the somewhat surprising inclusion 𝐵𝑉1 ⊆ 𝐶1 , although the analogous in­ clusion 𝐵𝑉 ⊆ 𝐶 is of course not true.

5.3 Spaces of differentiable functions |

359

which shows that 𝑔 ∈ 𝐵𝑉1 ([𝑎, 𝑏]). On the other hand, assume that 𝑓󸀠 vanishes some­ where in [𝑎, 𝑏], say 𝑓󸀠 (𝑎𝑘 ) = 0. Then sup |𝑓󸀠 (𝑡𝑗 )| = sup |𝑓󸀠 (𝑡𝑗 ) − 𝑓󸀠 (𝑎𝑘 )| ≤ Var(𝑓󸀠 ; [𝑎𝑘 , 𝑏𝑘 ]) .

𝑗=0,...,𝑚

𝑗=0,...,𝑚

Consequently, passing to the supremum over 𝑃 ∈ P([𝑎𝑘 , 𝑏𝑘 ]), we get Var(𝑔󸀠 ; [𝑎𝑘 , 𝑏𝑘 ]) ≤ Var(𝑓󸀠 ; [𝑎𝑘 , 𝑏𝑘 ]) [Var(ℎ󸀠 ; [𝑐, 𝑑]) + sup |ℎ󸀠 (𝑢)|] , 𝑐≤𝑢≤𝑑

and summing up over 𝑘 on both sides yields Var(𝑔󸀠 ; [𝑎, 𝑏]) ≤ Var(𝑓󸀠 ; [𝑎, 𝑏]) [Var(ℎ󸀠 ; [𝑐, 𝑑]) + sup |ℎ󸀠 (𝑢)|] .

(5.64)

𝑐≤𝑢≤𝑑

It remains to show that the operator 𝐶ℎ is bounded under the assumptions of The­ orem 5.35. To this end, we use the norm ⦀𝑓⦀𝐵𝑉1 := |𝑓(𝑎)| + ⦀𝑓󸀠 ⦀𝐵𝑉 = |𝑓(𝑎)| + ‖𝑓󸀠 ‖∞ + Var(𝑓󸀠 ; [𝑎, 𝑏]) which is equivalent to the norm (5.57). In the first case, from (5.63), we get Var(𝑔󸀠 ; [𝑎, 𝑏]) ≤ (‖𝑓󸀠 ‖𝐶 + Var(𝑓󸀠 ; [𝑎, 𝑏])) ( sup |ℎ󸀠 (𝑢)| + Var(ℎ󸀠 ; [𝑐, 𝑑])) , 𝑐≤𝑢≤𝑑

and hence ⦀𝑔⦀𝐵𝑉1 ≤ 2⦀𝑓⦀𝐵𝑉1 ⦀ℎ⦀𝐵𝑉1 . The reasoning in the second case is similar and follows from (5.64). Before stating a parallel result for the space 𝐴𝐶1 of functions with absolutely contin­ uous derivatives, we make some general remarks on absolutely continuous functions. If 𝑓 ∈ 𝐴𝐶1 ([𝑎, 𝑏]) and 𝑔 ∈ 𝐴𝐶loc (ℝ), we remark that 𝑔 ∘ 𝑓 need not be absolutely con­ tinuous. We give an example [75] involving the oscillation functions (0.86). Example 5.36. Let 𝑓𝛼,𝛽 : [0, 1] → ℝ be defined by (0.86), and let 𝑔 : ℝ → ℝ be defined by 𝑔(𝑢) := |𝑢|𝛾 . Here, we choose 𝛼, 𝛽 and 𝛾 in such a way that¹³ 0 1 was completely solved in [56], see Theo­ rem 5.56 in Section 5.6 below. Now, we solve the COP for the space 𝐿𝑖𝑝1𝛼 , where for simplicity, we restrict our­ selves to the case 𝛼 = 1. Again, the following theorem is in sharp contrast to Theo­ rem 5.10. Theorem 5.39. The operator (5.1) maps the space 𝐿𝑖𝑝1 ([𝑎, 𝑏]) into itself if and only if ℎ ∈ 𝐿𝑖𝑝1loc (ℝ). Moreover, in this case, the operator (5.1) is automatically bounded in the norm (5.56). Proof. Again, without loss of generality, we may assume that [𝑎, 𝑏] = [0, 1]. The ne­ cessity of the condition ℎ ∈ 𝐿𝑖𝑝1loc (ℝ) for the inclusion 𝐶ℎ (𝐿𝑖𝑝1 ) ⊆ 𝐿𝑖𝑝1 follows as before. Suppose that ℎ ∈ 𝐿𝑖𝑝1loc (ℝ), and let 𝑓 ∈ 𝐿𝑖𝑝1 ([0, 1]) and 𝑔 := 𝐶ℎ 𝑓, and thus 𝑔󸀠 = (ℎ󸀠 ∘ 𝑓)𝑓󸀠 . From ℎ ∈ 𝐿𝑖𝑝1loc (ℝ), we get 𝐶ℎ󸀠 (𝐿𝑖𝑝) ⊆ 𝐿𝑖𝑝, by Theorem 5.10. Combin­ ing this with 𝑓󸀠 ∈ 𝐿𝑖𝑝([0, 1]), we conclude that 𝑔󸀠 ∈ 𝐿𝑖𝑝([0, 1]) since 𝐿𝑖𝑝([0, 1]) is an algebra. So, we have shown that 𝑔 ∈ 𝐿𝑖𝑝1 ([0, 1]), i.e. 𝐶ℎ (𝐿𝑖𝑝1 ) ⊆ 𝐿𝑖𝑝1 . Now, we show that the operator 𝐶ℎ is bounded in the norm (5.56) under the as­ sumption ℎ ∈ 𝐿𝑖𝑝1loc (ℝ). Given 𝑓 ∈ 𝐿𝑖𝑝1 ([0, 1]) with ‖𝑓‖𝐿𝑖𝑝1 ≤ 𝑟 and using again (5.38), we have ‖𝑓󸀠 ‖𝐶 ≤ |𝑓󸀠 (0)| + 𝑙𝑖𝑝(𝑓󸀠 ) = ‖𝑓󸀠 ‖𝐿𝑖𝑝 ≤ 𝑟, ‖ℎ󸀠 ∘ 𝑓‖𝐶 ≤ 𝑘̃ 1 (𝑟) .

362 | 5 Nonlinear composition operators Moreover, since the operator 𝐶ℎ󸀠 is bounded in the space 𝐿𝑖𝑝([0, 1]), by Theo­ rem 5.24, we have 𝑙𝑖𝑝(ℎ󸀠 ∘ 𝑓) ≤ ‖𝐶ℎ󸀠 𝑓‖𝐿𝑖𝑝 , which implies that 𝐶ℎ is bounded in the norm (5.56). Observe that again we did not claim automatic continuity of 𝐶ℎ in Theorem 5.39. How­ ever, we are able to present a counterexample which shows that the operator (5.1) may map the space 𝐿𝑖𝑝1 into itself without being continuous in the norm (5.56). This coun­ terexample imitates Example 5.25. Example 5.40. Define ℎ : ℝ → ℝ by 0 { { {1 2 ℎ(𝑢) := { 2 𝑢 { { 1 {𝑢 − 2

for 𝑢 ≤ 0 , for 0 < 𝑢 < 1 , for 𝑢 ≥ 1 .

Since ℎ󸀠 (𝑢) ≡ 0 for 𝑢 ≤ 0 and ℎ󸀠 (𝑢) = min {𝑢, 1} for 𝑢 > 0, we certainly have ℎ ∈ (even ℎ ∈ 𝐿𝑖𝑝1 (ℝ)), and so 𝐶ℎ maps 𝐿𝑖𝑝1 ([0, 1]) into itself and is bounded, by Theorem 5.39. However, 𝐶ℎ is not continuous in the norm (5.56). To see this, consider the functions 𝑓, 𝑓𝑛 : [0, 1] → ℝ defined by 𝐿𝑖𝑝1loc (ℝ)

𝑓(𝑥) := 𝑥,

𝑓𝑛 (𝑥) :=

𝑛+1 𝑥 (0 ≤ 𝑥 ≤ 1) . 𝑛

Clearly, ‖𝑓𝑛 −𝑓‖𝐿𝑖𝑝1 = 𝑛1 → 0 as 𝑛 → ∞. On the other hand, the function 𝑔𝑛 defined by (5.58) and its derivative satisfy { 2𝑛+1 𝑥2 for 0 ≤ 𝑥 ≤ 𝜏𝑛 , 2𝑛2 𝑔𝑛 (𝑥) = { 𝑛+1 1 2 𝑥 − 2 (1 + 𝑥 ) for 𝜏𝑛 < 𝑥 ≤ 1 , { 𝑛 and { 2𝑛+1 𝑥 for 0 ≤ 𝑥 ≤ 𝜏𝑛 , 𝑛2 𝑔𝑛󸀠 (𝑥) = { 𝑛+1 − 𝑥 for 𝜏𝑛 < 𝑥 ≤ 1 , { 𝑛 𝑛 where 𝜏𝑛 := 𝑛+1 . Consequently, ‖𝐶ℎ 𝑓𝑛 − 𝐶ℎ 𝑓‖𝐿𝑖𝑝1 ≥ 𝑙𝑖𝑝(𝑔𝑛󸀠 ) ≥

|𝑔𝑛󸀠 (1) − 𝑔𝑛󸀠 (𝜏𝑛 )| 𝑛 + 1 ≡ 1, = 1 − 𝜏𝑛 𝑛+1

which shows that ‖𝐶ℎ 𝑓𝑛 − 𝐶ℎ 𝑓‖𝐿𝑖𝑝1 󴀀󴀂󴀠 0 as 𝑛 → ∞.

♥

Observe that the function ℎ in Example 5.40 belongs to 𝐿𝑖𝑝1loc (ℝ), and so also to 𝐶1 (ℝ), but has no second derivative at 0 and 1. This is not accidental, as the following theorem shows which provides a necessary and sufficient continuity condition and is parallel to Theorem 5.26. Theorem 5.41. The operator (5.1) maps the space 𝐿𝑖𝑝1 ([𝑎, 𝑏]) into itself and is continu­ ous in the norm (5.56) if and only if ℎ ∈ 𝐶2 (ℝ).

5.3 Spaces of differentiable functions

| 363

Since the proof of this theorem is quite similar to that of Theorem 5.26, we leave it as an exercise to the reader (Exercise 5.24). Our previous discussion shows that in all spaces under consideration, we get boundedness of the operator 𝐶ℎ for free, while continuity is a delicate problem: in the largest space 𝐶1 in (5.59), continuity holds, in the smallest space 𝐿𝑖𝑝1 in (5.59), con­ tinuity fails, and in the intermediate spaces 𝑅𝐵𝑉1𝑝 and 𝐴𝐶1 in (5.59), we do not know the answer. For the reader’s ease, we summarize our results in the following synoptic tables which complement the Tables 5.2–5.8 above. Table 5.9. The operator 𝐶ℎ in 𝐶1 ([𝑎, 𝑏]). 𝐶ℎ bounded in 𝐶1

𝐶ℎ (𝐶1 ) ⊆ 𝐶1 ⇕ ℎ ∈ 𝐶1 (ℝ)

⇔

⇔

𝐶ℎ continuous in 𝐶1

Table 5.10. The operator 𝐶ℎ in 𝐿𝑖𝑝1𝛼 ([𝑎, 𝑏]) (0 < 𝛼 ≤ 1). 𝐶ℎ bounded in 𝐿𝑖𝑝1𝛼

⇔

𝐶ℎ (𝐿𝑖𝑝1𝛼 ) ⊆ 𝐿𝑖𝑝1𝛼 ⇕ ℎ ∈ 𝐿𝑖𝑝1loc (ℝ)

⇐ ⇐

𝐶ℎ continuous in 𝐿𝑖𝑝1𝛼 ⇕ ℎ ∈ 𝐶2 (ℝ)

Table 5.11. The operator 𝐶ℎ in 𝐵𝑉1 ([𝑎, 𝑏]). 𝐶ℎ bounded in 𝐵𝑉1

⇔

𝐶ℎ (𝐵𝑉1 ) ⊆ 𝐵𝑉1 ⇕ ℎ ∈ 𝐵𝑉1loc (ℝ)

⇐

𝐶ℎ continuous in 𝐵𝑉1

⇐

𝐶ℎ continuous in 𝐴𝐶1

Table 5.12. The operator 𝐶ℎ in 𝐴𝐶1 ([𝑎, 𝑏]). 𝐶ℎ bounded in 𝐴𝐶1

⇔

𝐶ℎ (𝐴𝐶1 ) ⊆ 𝐴𝐶1 ⇕ ℎ ∈ 𝐴𝐶1loc (ℝ)

Table 5.13. The operator 𝐶ℎ in 𝑅𝐵𝑉1𝑝 ([𝑎, 𝑏]) (𝑝 > 1). 𝐶ℎ bounded in 𝑅𝐵𝑉1𝑝

⇔

𝐶ℎ (𝑅𝐵𝑉1𝑝 ) ⊆ 𝑅𝐵𝑉1𝑝 ⇕ ℎ ∈ 𝑅𝐵𝑉1𝑝,𝑙𝑜𝑐 (ℝ)

⇐

𝐶ℎ continuous in 𝑅𝐵𝑉1𝑝

Observe that the Tables 5.11–5.13 do not contain conditions on ℎ, both necessary and sufficient, under which the operator 𝐶ℎ is continuous in the norm of the corresponding space. As far as we know, such continuity criteria are not known. Of course, one may again try to find conditions which are just sufficient (Exercises 5.21–5.23).

364 | 5 Nonlinear composition operators

5.4 Global Lipschitz continuity Suppose that the composition operator 𝐶ℎ given by (5.1) maps a normed space 𝑋 into a normed space 𝑌. If we want to apply fixed point theorems to nonlinear problems in in­ finite dimensional Banach spaces, just continuity of the operators involved is too weak to guarantee the existence of fixed points. In case of the Schauder fixed point princi­ ple, we need some compactness requirement which in infinite dimensional spaces is often quite restrictive. On the other hand, in case of the Banach–Caccioppoli fixed point principle, one usually imposes a (global) Lipschitz condition of the type ‖𝐶ℎ 𝑓 − 𝐶ℎ 𝑔‖𝑋 ≤ 𝐾‖𝑓 − 𝑔‖𝑋

(𝑓, 𝑔 ∈ 𝑋)

(5.67)

with 𝐾 < 1 to guarantee the existence (and uniqueness) of a fixed point of 𝐶ℎ in 𝑋. Unfortunately, if 𝐶ℎ is the nonlinear composition operator (5.1) (or, more generally, the nonlinear superposition operator (5.2), the global Lipschitz condition (5.67) leads to a strong degeneracy for the generating function ℎ in many function spaces 𝑋. In fact, from (5.67), it often follows that the function ℎ must be affine, which means that ℎ(𝑢) = 𝛼 + 𝛽𝑢 (𝛼, 𝛽 ≥ 0) (5.68) in case of the composition operator (5.1), or ℎ(𝑡, 𝑢) = 𝛼(𝑡) + 𝛽(𝑡)𝑢

(𝛼, 𝛽 ∈ 𝑌)

(5.69)

in case of the superposition operator (5.2). This was shown, even in the nonau­ tonomous case of the operator (5.2), for – the space 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) of Hölder continuous functions of order 𝛼 < 1 with norm (0.71) in [193]; – the space 𝐶𝑛 ([𝑎, 𝑏]) of 𝑛-times continuously differentiable functions with norm (0.63) in [194]; – the space 𝑊𝐵𝑉2,𝑝 ([𝑎, 𝑏]) of functions of bounded (2, 𝑝)-variation with norm (2.143) in [204]; – the space 𝐿𝑖𝑝𝑛 ([𝑎, 𝑏]) of functions with Lipschitz continuous 𝑛-th derivative with norm (0.78) in [162]; – the space 𝐿𝑖𝑝𝑛𝛼 ([𝑎, 𝑏]) of functions with Hölder continuous 𝑛-th derivative in [187]; – the space 𝐴𝐶𝑛 ([𝑎, 𝑏]) of functions with absolutely continuous 𝑛-th derivative in [289]; – the space 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) of functions of bounded Riesz variation for 1 < 𝑝 < ∞ with norm (2.90) in [205]¹⁴ and [224]; and – the space 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) of functions of bounded Riesz–Medvedev variation with norm (2.99) by [217].

14 To be precise, in [205] the authors consider the Sobolev space 𝑊𝑝1 of functions with distributional first derivative in 𝐿 𝑝 ; as we have seen in Theorem 3.34, this is the same as 𝑅𝐵𝑉𝑝 .

5.4 Global Lipschitz continuity

| 365

Since Janusz Matkowski was the first, to the best of our knowledge, to discover this degeneracy phenomenon, we will use the following terminology. Definition 5.42. We say that a pair (𝑋, 𝑌) of two normed spaces (𝑋, ‖⋅‖) and (𝑌, ‖⋅‖) has the Matkowski property if whenever the operator (5.1) [the operator (5.2), respectively] maps the space 𝑋 into the space 𝑌 and satisfies (5.67) [(5.67) with 𝐶ℎ replaced by 𝑆ℎ , respectively], the corresponding function ℎ must have the form (5.68) [the form (5.69), respectively]. In case 𝑋 = 𝑌, we simply say that 𝑋 has the Matkowski property. ◼ Thus, our collection above shows that many spaces occurring frequently in applica­ tions have the Matkowski property. We point out, however, that there are important function spaces which do not have the Matkowski property. Two examples of such spaces are contained in the following Theorems 5.43 and 5.44; we state them for the general operator (5.2). Theorem 5.43. Suppose that the operator 𝑆ℎ given by (5.2) maps the space 𝐶([𝑎, 𝑏]) into itself. Then 𝑆ℎ satisfies (5.67) if and only if the corresponding function ℎ satisfies |ℎ(𝑡, 𝑢) − ℎ(𝑡, 𝑣)| ≤ 𝑔(𝑡)|𝑢 − 𝑣|

(𝑎 ≤ 𝑡 ≤ 𝑏, 𝑢, 𝑣 ∈ ℝ)

(5.70)

for some continuous function 𝑔 : [𝑎, 𝑏] → ℝ. Theorem 5.44. Suppose that the operator 𝑆ℎ given by (5.2) maps the space 𝐿 𝑝 ([𝑎, 𝑏]) into the space 𝐿 𝑞 ([𝑎, 𝑏]), where 1 ≤ 𝑞 ≤ 𝑝 < ∞. Then 𝑆ℎ satisfies (5.67) if and only if the corresponding function ℎ satisfies (5.70), where 𝑔 ∈ 𝐿 𝑝𝑞/(𝑝−𝑞) ([𝑎, 𝑏]). The proof of Theorems 5.43 and 5.44 may be found in [20]. We remark that necessary and sufficient conditions under which the superposition operator (5.2) fulfills the hy­ potheses of these theorems are well known. Clearly, condition (5.70) may be replaced by the simpler condition |ℎ(𝑡, 𝑢) − ℎ(𝑡, 𝑣)| ≤ 𝑘|𝑢 − 𝑣| (𝑎 ≤ 𝑡 ≤ 𝑏, 𝑢, 𝑣 ∈ ℝ) ,

(5.71)

where 𝑘 := ‖𝑔‖𝐶 , i.e. by a global Lipschitz condition for the function 𝑓(𝑡, ⋅) on ℝ. In case of the autonomous composition operator (5.1), condition (5.71) becomes |ℎ(𝑢) − ℎ(𝑣)| ≤ 𝑘|𝑢 − 𝑣|

(𝑢, 𝑣 ∈ ℝ) ,

(5.72)

which is a global Lipschitz condition for 𝑓 on the whole real axis. Of course, (5.72) is much more restrictive than the local condition (5.13). We now prove a general theorem which provides a unified approach to spaces with the Matkowski property for the autonomous operator (5.1). Without loss of generality, we take [𝑎, 𝑏] = [0, 1]. We consider the space 𝑃𝑛 ([0, 1]) of all polynomials of degree ≤ 𝑛

366 | 5 Nonlinear composition operators on [0, 1] equipped with the 𝐶𝑛 -norm (0.63). In particular, on the space 𝑃1 ([0, 1]) of all affine functions 𝑓(𝑥) = 𝑐𝑥+𝑑, we consider the 𝐶1 -norm ‖𝑓‖𝐶1 = |𝑐|+|𝑑|. It is interesting to note that the global Lipschitz condition (5.67) for 𝐶ℎ may even be weakened: just uniform continuity¹⁵ of 𝐶ℎ suffices to imply (5.68). Theorem 5.45. Suppose that the operator (5.1) maps a normed space 𝑋 into a normed space 𝑌 and is uniformly continuous on 𝑋. Assume that the space 𝑃1 ([0, 1]) of affine func­ tions equipped with the 𝐶1 norm (0.65) is imbedded into 𝑋, and 𝑌 is imbedded into the Hölder space 𝐿𝑖𝑝𝛾 ([0, 1]) for some 𝛾 ∈ (0, 1] with norm (0.71). Then there exist constants 𝛼, 𝛽 ∈ ℝ such that (5.68) holds true. Proof. From our assumptions, it follows that we can find a 𝛿 > 0 such that ‖𝐶ℎ 𝑓 − 𝐶ℎ 𝑔‖𝐿𝑖𝑝𝛾 ≤ 1 for all 𝑓, 𝑔 ∈ 𝑃1 ([0, 1]) satisfying ‖𝑓 − 𝑔‖𝐶1 ≤ 𝛿. Fix 𝜔 > 0 and 𝑣 ∈ [−𝛿, 𝛿], and define 𝑓𝜔 , 𝑔𝜔 ∈ 𝑃1 ([0, 1]) by 𝑓𝜔 (𝑡) := 𝜔𝑡 + 𝑣 and 𝑔𝜔 (𝑡) := 𝜔𝑡. Since ‖𝑓𝜔 − 𝑔𝜔 ‖𝐶1 = |𝑣| ≤ 𝛿, we know that 𝑙𝑖𝑝𝛾 (𝐶ℎ 𝑓𝜔 − 𝐶ℎ 𝑔𝜔 ) ≤ 1, and hence |ℎ(𝜔𝑠 + 𝑣) − ℎ(𝜔𝑠) − ℎ(𝜔𝑡 + 𝑣) + ℎ(𝜔𝑡)| ≤ |𝑠 − 𝑡|𝛾 . Putting, in particular, 𝑠 = 𝑢/𝜔 and 𝑡 = 0, we conclude that 󵄨󵄨 𝑢 󵄨󵄨𝛾 |ℎ(𝑢 + 𝑣) − ℎ(𝑢) − ℎ(𝑣) + ℎ(0)| ≤ 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 → 0 󵄨𝜔󵄨

(𝜔 → ∞) .

Suppose first that ℎ(0) = 0. Then the last equality shows that ℎ(𝑢 + 𝑣) = ℎ(𝑢) + ℎ(𝑣)

(𝑢, 𝑣 ∈ ℝ, |𝑣| ≤ 𝛿)

which by standard arguments, implies that ℎ(𝑢) = 𝛽𝑢 with 𝛽 = ℎ(1). Replacing ℎ in case ℎ(0) ≠ 0 by the function 𝑢 󳨃→ ℎ(𝑢) − ℎ(0) the statement follows with 𝛼 = ℎ(1) − ℎ(0). The range of applicability of Theorem 5.45 seems to be rather limited because the hy­ pothesis 𝑌 󳨅→ 𝐿𝑖𝑝𝛾 ([0, 1]) is quite strong. However, some information may be deduced from this theorem which we state as Corollary 5.46. The operator (5.1) is uniformly continuous on either of the spaces 𝐿𝑖𝑝, 𝐿𝑖𝑝𝛼 , 𝐶1 , 𝐴𝐶1 , 𝐵𝑉1 , 𝐿𝑖𝑝1 , or 𝑅𝐵𝑉1𝑝 if and only if the corresponding function ℎ has the form (5.68). To cover a larger family of spaces with the Matkowski property, we use the space 𝑊𝐵𝑉𝑝 of bounded 𝑝-variation in Wiener’s sense (in particular, the classical space 𝑊𝐵𝑉1 = 𝐵𝑉).

15 Here, we mean uniform continuity on the whole space; this has to be carefully distinguished from uniform continuity on bounded subsets which we considered in Theorem 5.26. Also, note that (5.68) trivially implies that the corresponding operator 𝐶ℎ is uniformly continuous on 𝑋.

5.4 Global Lipschitz continuity

| 367

Theorem 5.47. Assume that the space 𝑋 contains all piecewise linear functions, and the space 𝑌 is imbedded into the space 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) for some 𝑝 ≥ 1 with norm (1.65). Then the pair (𝑋, 𝑌) has the Matkowski property. Proof. Suppose that the composition operator (5.1) maps 𝑋 into 𝑌 and satisfies the Lipschitz condition (5.67). From our hypothesis 𝑌 󳨅→ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]), it then follows that (5.73) ‖𝐶ℎ 𝑓 − 𝐶ℎ 𝑔‖𝑊𝐵𝑉𝑝 ≤ 𝐿‖𝑓 − 𝑔‖𝑋 (𝑓, 𝑔 ∈ 𝑋) for some 𝐿 > 0. Let 𝑎 ≤ 𝑠 < 𝑡 ≤ 𝑏, and let 𝑃𝑚 := {𝑡0 , 𝑡1 , . . . , 𝑡2𝑚 } ∈ P([𝑠, 𝑡]) be the equidistant partition defined by 𝑡0 = 𝑠,

𝑡𝑗 − 𝑡𝑗−1 =

𝑡−𝑠 2𝑚

(𝑗 = 1, 2, . . . , 2𝑚) .

Given 𝑢, 𝑣 ∈ ℝ with 𝑢 ≠ 𝑣, define 𝑓, 𝑔 : [𝑎, 𝑏] → ℝ by 𝑣 { { { 𝑓(𝑥) := { 𝑢+𝑣 { { 2 {linear and

𝑢+𝑣

{ { { 2 𝑔(𝑥) := {𝑢 { { {linear

if 𝑥 = 𝑡𝑗 for some even 𝑗 , if 𝑥 = 𝑡𝑗 for some odd 𝑗 , otherwise ,

if 𝑥 = 𝑡𝑗 for some even 𝑗 , if 𝑥 = 𝑡𝑗 for some odd 𝑗 , otherwise ,

respectively. Then the difference 𝑓 − 𝑔 trivially satisfies |𝑓(𝑥) − 𝑔(𝑥)| ≡

|𝑢 − 𝑣| 2

(𝑎 ≤ 𝑥 ≤ 𝑏) .

Consequently, substituting these functions 𝑓 and 𝑔 into (5.73) yields ‖𝐶ℎ 𝑓 − 𝐶ℎ 𝑔‖𝑊𝐵𝑉𝑝 ≤

𝐾|𝑢 − 𝑣| , 2

where the constant 𝐾 is given by 𝐾 := 𝐿‖𝑓1 ‖𝑋 and 𝑓1 denotes the constant function 𝑓(𝑥) ≡ 1. In particular, for the partition 𝑃𝑚 , as above, we have 𝑚 󵄨 󵄨𝑝 ∑ 󵄨󵄨󵄨󵄨(ℎ ∘ 𝑓)(𝑡2𝑗 ) − (ℎ ∘ 𝑔)(𝑡2𝑗 ) − (ℎ ∘ 𝑓)(𝑡2𝑗−1 ) + (ℎ ∘ 𝑔)(𝑡2𝑗−1 )󵄨󵄨󵄨󵄨

𝑗=1

𝐾𝑝 |𝑢 − 𝑣|𝑝 = Var𝑊 . 𝑝 (𝐶ℎ 𝑓 − 𝐶ℎ 𝑔, 𝑃𝑚 ; [𝑠, 𝑡]) ≤ 2𝑝 However, by definition of the functions 𝑓 and 𝑔, we have (ℎ ∘ 𝑓)(𝑡2𝑗 ) − (ℎ ∘ 𝑔)(𝑡2𝑗 ) − (ℎ ∘ 𝑓)(𝑡2𝑗−1 ) + (ℎ ∘ 𝑔)(𝑡2𝑗−1 ) = ℎ(𝑣) − ℎ( 𝑢+𝑣 ) − ℎ( 𝑢+𝑣 ) + ℎ(𝑢) 2 2

(5.74)

368 | 5 Nonlinear composition operators which is actually independent of 𝑚. So, combining this with (5.74), we obtain 𝑚

∑ |ℎ(𝑣) − ℎ( 𝑢+𝑣 ) − ℎ( 𝑢+𝑣 ) + ℎ(𝑢)|𝑝 ≤ 2 2

𝑗=1

𝐾𝑝 |𝑢−𝑣|𝑝 2𝑝

.

(5.75)

Letting now 𝑚 → ∞ in (5.75), we conclude that the function ℎ satisfies the Cauchy functional equation ℎ(𝑢) + ℎ(𝑣) = 2ℎ( 𝑢+𝑣 ). 2 Since ℎ is continuous, it follows that ℎ(𝑢) = 𝛼 + 𝛽𝑢 with 𝛽 = ℎ(0) and 𝛼 = ℎ(1)− ℎ(0) as desired. In Chapter 6, we shall obtain a far reaching generalization of Theorem 5.47 for the case of the superposition operator (5.2). Theorem 5.47 applies to the composition operator (5.1) in any space 𝑋 which is continuously imbedded into 𝑊𝐵𝑉𝑝 for some 𝑝 ≥ 1, such as 𝑊𝐵𝑉𝑝 itself, 𝐵𝑉, 𝑅𝐵𝑉𝑝 , or 𝐴𝐶. Moreover, Proposition 2.33 shows that Theorem 5.47 applies as well to the Waterman space Λ 𝑞 𝐵𝑉 for any 𝑞 ∈ (0, 1) since Λ 𝑞 𝐵𝑉 󳨅→ 𝑊𝐵𝑉𝑝 for 𝑝 ≥ 1/(1 − 𝑞).

5.5 Local Lipschitz continuity As we have seen, imposing the global Lipschitz condition (5.67) to the composition operator 𝐶ℎ in 𝐵𝑉 leads to a very restrictive condition for the generating function ℎ. So, the question arises regarding how to replace (5.67) by some milder condition for 𝐶ℎ which does not lead to such a narrow class of generating functions ℎ. A good idea is to replace the global Lipschitz condition (5.67) by a local Lipschitz condition of type ‖𝐶ℎ 𝑓 − 𝐶ℎ 𝑔‖𝑋 ≤ 𝐾(𝑟)‖𝑓 − 𝑔‖𝑋

(𝑓, 𝑔 ∈ 𝑋; ‖𝑓‖, ‖𝑔‖ ≤ 𝑟) ,

(5.76)

i.e. imposing Lipschitz conditions only on closed balls of radius 𝑟 > 0 and allowing the corresponding Lipschitz constant 𝐾(𝑟) to depend on 𝑟 (and, as a matter of fact, tending to infinity as 𝑟 → ∞). It turns out that the local condition (5.76) is much more reasonable than the global condition (5.67) for 𝐶ℎ , insofar as it does not lead to a strong degeneracy for the generating function ℎ. We will show this now for the classical space 𝐵𝑉([𝑎, 𝑏]) in case of the composition operator (5.1). First, we need a technical lemma which seems to be of some interest on its own. Apart from condition (5.76) for the operator 𝐶ℎ , at many places, we will need the local Lipschitz condition (5.13) for the function ℎ as well as the local Lipschitz condition (5.14) for its derivative. ̃ Moreover, we will use the characteristics 𝑘(𝑟) and 𝑘̃ 1 (𝑟) defined in (5.15) and (5.16), respectively.

369

5.5 Local Lipschitz continuity |

Lemma 5.48. Suppose that the derivative of a function ℎ ∈ 𝐶1 (ℝ) satisfies the local Lipschitz condition (5.14). Then for |𝑥1 |, |𝑥2 |, |𝑦1 |, |𝑦2 | ≤ 𝑟, we have the estimate |ℎ(𝑥1 ) − ℎ(𝑦1 ) − ℎ(𝑥2 ) + ℎ(𝑦2 )| ≤ 𝑘1 (𝑟) (|𝑥1 − 𝑥2 | + |𝑦1 − 𝑦2 |) (|𝑥1 − 𝑦1 | + |𝑥2 − 𝑦2 |) + 𝑘̃ 1 (𝑟)|𝑥1 − 𝑦1 − 𝑥2 + 𝑦2 | . Proof. We distinguish the cases |𝑥1 − 𝑦1 | + |𝑥2 − 𝑦2 | ≤ |𝑥1 − 𝑥2 | + |𝑦1 − 𝑦2 |

(5.77)

|𝑥1 − 𝑦1 | + |𝑥2 − 𝑦2 | > |𝑥1 − 𝑥2 | + |𝑦1 − 𝑦2 | .

(5.78)

and In the first case, we choose, by the mean value theorem, some 𝜉𝑖 between 𝑥𝑖 and 𝑦𝑖 (hence, |𝜉𝑖 | ≤ 𝑟) with ℎ(𝑥𝑖 ) − ℎ(𝑦𝑖 ) = ℎ󸀠 (𝜉𝑖 )(𝑥𝑖 − 𝑦𝑖 ) (𝑖 = 1, 2) . Using (5.77), a straightforward but cumbersome case distinction shows that |𝜉1 − 𝜉2 | ≤ |𝑥1 − 𝑥2 | + |𝑦1 − 𝑦2 | . Consequently, |ℎ(𝑥1 ) − ℎ(𝑦1 ) − ℎ(𝑥2 ) + ℎ(𝑦2 )| = |ℎ󸀠 (𝜉1 )(𝑥1 − 𝑦1 ) − ℎ󸀠 (𝜉2 )(𝑥2 − 𝑦2 )| = | [ℎ󸀠 (𝜉1 ) − ℎ󸀠 (𝜉2 )] (𝑥1 − 𝑦1 ) + ℎ󸀠 (𝜉2 ) (𝑥1 − 𝑦1 − 𝑥2 + 𝑦2 ) |

(5.79)

≤ 𝑘1 (𝑟) (|𝑥1 − 𝑥2 | + |𝑦1 − 𝑦2 |) |𝑥1 − 𝑦1 | + 𝑘̃ 1 (𝑟)|𝑥1 − 𝑦2 − 𝑥2 + 𝑦2 | . In the second case, we choose, again by the mean value theorem, some 𝜂𝑥 between 𝑥1 and 𝑥2 and some 𝜂𝑦 between 𝑦1 and 𝑦2 (hence, |𝜂𝑥 |, |𝜂𝑦 | ≤ 𝑟) satisfying ℎ(𝑥1 ) − ℎ(𝑥2 ) = ℎ󸀠 (𝜂𝑥 )(𝑥1 − 𝑥2 ),

ℎ(𝑦1 ) − ℎ(𝑦2 ) = ℎ󸀠 (𝜂𝑦 )(𝑦1 − 𝑦2 ) .

As before, a straightforward but cumbersome case distinction, now building on (5.78), shows that |𝜂𝑥 − 𝜂𝑦 | ≤ |𝑥1 − 𝑦1 | + |𝑥2 − 𝑦2 | . So, in this case, we obtain |ℎ(𝑥1 ) − ℎ(𝑦1 ) − ℎ(𝑥2 ) + ℎ(𝑦2 )| = |ℎ󸀠 (𝜂𝑥 )(𝑥1 − 𝑥2 ) − ℎ󸀠 (𝜂𝑦 )(𝑦1 − 𝑦2 )| = | [ℎ󸀠 (𝜂𝑥 ) − ℎ󸀠 (𝜂𝑦 )] (𝑥1 − 𝑥2 ) + ℎ󸀠 (𝜂𝑦 ) (𝑥1 − 𝑦1 − 𝑥2 + 𝑦2 ) |

(5.80)

≤ 𝑘1 (𝑟) (|𝑥1 − 𝑦1 | + |𝑥2 − 𝑦2 |) |𝑥1 − 𝑥2 | + 𝑘̃ 1 (𝑟)|𝑥1 − 𝑦1 − 𝑥2 + 𝑦2 | . Estimating the terms after 𝑘1 (𝑟) in (5.79) and (5.80) in a unified manner, we obtain the statement.

370 | 5 Nonlinear composition operators Theorem 5.49. Suppose that the operator (5.1) maps the space 𝐵𝑉([𝑎, 𝑏]) into itself and is continuous and bounded with respect to the norm (1.16). Then the operator (5.1) sat­ isfies (5.76) if and only if ℎ󸀠 exists on ℝ and satisfies (5.14). Proof. By the COP-invariance of the space 𝐵𝑉 (Example 5.3), there is no loss of gen­ erality to assume that [𝑎, 𝑏] = [0, 1]. Suppose first that the derivative ℎ󸀠 of ℎ satisfies the local Lipschitz condition (5.14), and define 𝑘̃ 1 (𝑟) as in (5.16). Fix 𝑓, 𝑔 ∈ 𝐵𝑉([0, 1]) with ‖𝑓‖𝐵𝑉 ≤ 𝑟 and ‖𝑔‖𝐵𝑉 ≤ 𝑟. Given a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]), we apply Lemma 5.48 to the choice 𝑓(𝑡𝑗 ) =: 𝑥1 ,

𝑓(𝑡𝑗−1 ) =: 𝑥2 ,

𝑔(𝑡𝑗 ) =: 𝑦1 ,

𝑔(𝑡𝑗−1 ) =: 𝑦2

(𝑗 = 1, 2, . . . , 𝑚) .

As a result, we get the estimate |ℎ(𝑓(𝑡𝑗 )) − ℎ(𝑔(𝑡𝑗 )) − ℎ(𝑓(𝑡𝑗−1 )) + ℎ(𝑔(𝑡𝑗−1 ))| ≤ 𝑘1 (𝑟) (|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| + |𝑔(𝑡𝑗 ) − 𝑔(𝑡𝑗−1 )|) (|𝑓(𝑡𝑗 ) − 𝑔(𝑡𝑗 )| + |𝑓(𝑡𝑗−1 ) − 𝑔(𝑡𝑗−1 )|) + 𝑘̃ 1 (𝑟)|𝑓(𝑡𝑗 ) − 𝑔(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 ) + 𝑔(𝑡𝑗−1 )| . Taking the sum over 𝑗 = 1, 2, . . . , 𝑚 and using the norm ‖⋅‖∞ from (0.39), we obtain 𝑚

∑ |(ℎ ∘ 𝑓)(𝑡𝑗 ) − (ℎ ∘ 𝑔)(𝑡𝑗 ) − (ℎ ∘ 𝑓)(𝑡𝑗−1 ) + (ℎ ∘ 𝑔)(𝑡𝑗−1 )|

𝑗=1

𝑚

≤ 2𝑘1 (𝑟)‖𝑓 − 𝑔‖∞ ∑ (|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| + |𝑔(𝑡𝑗 ) − 𝑔(𝑡𝑗−1 )|) 𝑗=1

𝑚

+ 𝑘̃ 1 (𝑟) ∑ |𝑓(𝑡𝑗 ) − 𝑔(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 ) + 𝑔(𝑡𝑗−1 )| 𝑗=1

≤ 2𝑘1 (𝑟)‖𝑓 − 𝑔‖∞ (Var(𝑓) + Var(𝑔)) + 𝑘̃ 1 (𝑟) Var(𝑓 − 𝑔) ≤ 𝑘̂ 1 (𝑟)‖𝑓 − 𝑔‖𝐵𝑉 with 𝑘̂ 1 (𝑟) := max {4𝑟𝑘1 (𝑟), 𝑘̃ 1 (𝑟)}. This proves the first part of the theorem. Now, we suppose that 𝐶ℎ maps the space 𝐵𝑉([0, 1]) into itself and satisfies a local Lipschitz condition (5.76) in the norm (1.16). By Theorem 5.9, ℎ then satisfies the local Lipschitz condition (5.13). We will only use the fact that ℎ is absolutely continuous on [−𝑟, 𝑟]. There is a nullset 𝑁 ⊂ ℝ such that ℎ󸀠 exists on [−𝑟, 𝑟] \ 𝑁. By Exercise 3.32, it suffices to show that the restriction function 𝑓󸀠 |[−𝑟,𝑟]\𝑁 to [−𝑟, 𝑟]\ 𝑁 satisfies a Lipschitz condition. We show this for the Lipschitz constant 𝐿 := 𝐾(3𝑟 + 1), where 𝐾 = 𝐾(𝑟) is the constant from (5.76). Thus, assume by contradiction that there are 𝑢0 , 𝑣0 ∈ [−𝑟, 𝑟] \ 𝑁 with |ℎ󸀠 (𝑢0 ) − ℎ󸀠 (𝑣0 )| > 𝐿|𝑢0 − 𝑣0 | . Let 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]) be any partition where 𝑚 := ent

1 ∈ℕ |𝑢0 − 𝑣0 |

5.5 Local Lipschitz continuity

| 371

denotes the integer part of 1/|𝑢0 − 𝑣0 |, and let 𝑓 : [0, 1] → ℝ be defined by 𝑢0 { { { 𝑓(𝑥) := {𝑣0 { { {linear

if 𝑥 = 𝑡𝑗 for some even 𝑗 if 𝑥 = 𝑡𝑗 for some odd 𝑗

(5.81)

otherwise .

Then 𝑓 has bounded variation, being even Lipschitz continuous on [0, 1], with Var(𝑓; [0, 1]) = Var(𝑓, 𝑃; [0, 1]) = 𝑚|𝑢0 − 𝑣0 | < 1 + 2𝑟 , and so ‖𝑓‖𝐵𝑉 < 1 + 3𝑟. Hence, if 𝑛 is sufficiently large, the function 𝑓𝑛 ∈ 𝐿𝑖𝑝([0, 1]) defined by 𝑓𝑛 (𝑡) := 𝑓(𝑡) + 1/𝑛 also satisfies ‖𝑓𝑛 ‖𝐵𝑉 < 1 + 3𝑟. By hypothesis, the function 𝑔𝑛 : [0, 1] → ℝ defined by 𝑔𝑛 := 𝑛 (𝐶ℎ 𝑓𝑛 − 𝐶ℎ 𝑓) thus satisfies ‖𝑔𝑛 ‖𝐵𝑉 = 𝑛‖𝐶ℎ 𝑓𝑛 − 𝐶ℎ 𝑓‖𝐵𝑉 ≤ 𝐿𝑛‖𝑓𝑛 − 𝑓‖𝐵𝑉 = 𝐿 for all sufficiently large 𝑛. In particular, 𝑚

𝐿 ≥ Var(𝑔𝑛 ; [0, 1])) ≥ ∑ |𝑔𝑛 (𝑡𝑗 ) − 𝑔𝑛 (𝑡𝑗−1 )| 𝑗=1

for these 𝑛. However, the definition of the derivative implies that {ℎ󸀠 (𝑢0 ) lim 𝑔𝑛 (𝑡𝑗 ) = { 󸀠 𝑛→∞ ℎ (𝑣0 ) {

if 𝑗 is even , if 𝑗 is odd ;

thus, we finally obtain 𝑚

𝐿 ≥ ∑ |ℎ󸀠 (𝑢0 ) − ℎ󸀠 (𝑣0 )| > 𝐿𝑚|𝑢0 − 𝑣0 | ≥ 𝐿 , 𝑗=1

that is, a contradiction. The proof is complete. Observe that the proof of Theorem 5.49 provides an interesting interconnection be­ tween the Lipschitz constant 𝑘1 (𝑟) of ℎ󸀠 in (5.14) and the Lipschitz constant 𝐾(𝑟) of 𝐶ℎ in (5.76): from (5.14), it follows that (5.76) holds with 𝐾(𝑟) ≤ max {4𝑟𝑘1 (𝑟), 𝑘̃ 1 (𝑟)} ,

(5.82)

where 𝑘̃ 1 (𝑟) is given by (5.16). Conversely, from (5.76), it follows that (5.14) holds with 𝑘1 (𝑟) ≤

2𝐾(2𝑟) + 1 . 𝑟

(5.83)

Such estimates are useful for applying fixed point theorems to problems involving nonlinear composition operators. Moreover, they sometimes make it possible to prove the degeneracy results from the previous section quite easily (see, e.g. Exercise 5.28). In the same way as we have obtained an essential extension of Theorem 5.9 in Theorem 5.10, we may give the following important generalization of Theorem 5.49:

372 | 5 Nonlinear composition operators Theorem 5.50. The following six statements about the operator (5.1) and the generating function ℎ : ℝ → ℝ are equivalent. (a) The operator (5.1) satisfies the local Lipschitz condition (5.76) between the spaces 𝐿𝑖𝑝([𝑎, 𝑏]) and 𝐵𝑉([𝑎, 𝑏]). (b) The operator (5.1) satisfies the local Lipschitz condition (5.76) in the space 𝐿𝑖𝑝([𝑎, 𝑏]). (c) The operator (5.1) satisfies the local Lipschitz condition (5.76) in the space 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]). (d) The operator (5.1) satisfies the local Lipschitz condition (5.76) in the space 𝐴𝐶([𝑎, 𝑏]). (e) The operator (5.1) satisfies the local Lipschitz condition (5.76) in the space 𝐵𝑉([𝑎, 𝑏]). (f) The derivative of the function ℎ exists and satisfies (5.14). Proof. We take again [𝑎, 𝑏] = [0, 1]. The inclusions (5.23) show that any of the state­ ments (b), (c), (d) or (e) implies (a). So, we first prove, as in the proof of Theorem 5.10, that (a) implies (f). To this end, we analyze again the proof of Theorem 5.49. If we take the partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]) equidistant, i.e. such that 𝑚(𝑡𝑗 − 𝑡𝑗−1 ) = 1, then the function 𝑓 defined by (5.81) is Lipschitz continuous with 𝑙𝑖𝑝(𝑓) = 𝑚|𝑢0 − 𝑣0 | < 1 + 2𝑟 , and so ‖𝑓‖𝐿𝑖𝑝 < 1 + 3𝑟. Hence, if 𝑛 is sufficiently large, the function 𝑓𝑛 ∈ 𝐿𝑖𝑝([0, 1]) defined by 𝑓𝑛 (𝑡) := 𝑓(𝑡) + 1/𝑛 also satisfies ‖𝑓𝑛 ‖𝐿𝑖𝑝 < 1 + 3𝑟. By hypothesis, the function 𝑔𝑛 : [0, 1] → ℝ defined by 𝑔𝑛 := 𝑛 (𝐶ℎ 𝑓𝑛 − 𝐶ℎ 𝑓) then satisfies ‖𝑔𝑛 ‖𝐵𝑉 = 𝑛‖𝐶ℎ 𝑓𝑛 − 𝐶ℎ 𝑓‖𝐵𝑉 ≤ 𝑛𝐿‖𝑓𝑛 − 𝑓‖𝐿𝑖𝑝 = 𝐿 , where 𝐿 = 𝐾(3𝑟 + 1) as before.¹⁶ The remaining part of the proof goes like that of Theorem 5.49. So, we conclude that (a) implies (f). Conversely, if the derivative of ℎ exists on ℝ and satisfies (5.14), then 𝐶ℎ satisfies (5.76) in the space 𝐵𝑉, by Theorem 5.49. However, all inclusions in (5.23) are indeed continuous imbeddings, and so 𝐶ℎ satisfies (5.76) also in the space 𝐿𝑖𝑝, 𝑅𝐵𝑉𝑝 , and 𝐴𝐶. This shows that (f) implies any of the other statements (a)–(e), and we are done. For the sake of completeness, we now prove that the same result holds in the Hölder space 𝐿𝑖𝑝𝛼 . Theorem 5.51. Suppose that the operator (5.1) maps the space 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) into itself and is continuous and bounded with respect to the norm (0.71). Then the operator (5.1) sat­ isfies (5.76) if and only if ℎ󸀠 exists on ℝ and satisfies (5.14). Proof. Suppose first that the derivative ℎ󸀠 of ℎ satisfies the local Lipschitz condition ̃ as in (5.15). Fix 𝑠, 𝑡 ∈ [𝑎, 𝑏] and 𝑓, 𝑔 ∈ 𝐿𝑖𝑝 ([𝑎, 𝑏]) with ‖𝑓‖ (5.14), and define 𝑘(𝑟) 𝐿𝑖𝑝𝛼 ≤ 𝛼

16 Here, we consider the Lipschitz condition (5.76) between different spaces: the norm on the left-hand side of (5.76) is taken in 𝐵𝑉([0, 1]), while the norm on the right-hand side of (5.76) is taken in 𝐿𝑖𝑝([0, 1]).

5.5 Local Lipschitz continuity

| 373

𝑟 and ‖𝑔‖𝐿𝑖𝑝𝛼 ≤ 𝑟. Then applying Lemma 5.48 to the choice 𝑓(𝑠) =: 𝑥1 , 𝑓(𝑡) =: 𝑥2 , 𝑔(𝑠) =: 𝑦1 , and 𝑔(𝑡) =: 𝑦2 (with 𝑠 ≠ 𝑡) yields |(ℎ ∘ 𝑓)(𝑠) − (ℎ ∘ 𝑔)(𝑠) − (ℎ ∘ 𝑓)(𝑡) + (ℎ ∘ 𝑔)(𝑡)| ̃ − 𝑔(𝑠) − 𝑓(𝑡) + 𝑔(𝑡)| . ≤ 2𝑘(𝑟) (|𝑓(𝑠) − 𝑓(𝑡)| + |𝑔(𝑠) − 𝑔(𝑡)|) ‖𝑓 − 𝑔‖𝐶 + 𝑘(𝑟)|𝑓(𝑠) Dividing by |𝑠 − 𝑡|𝛼 and passing to the supremum over 𝑠 ≠ 𝑡, we arrive at ̃ ℎ𝛼 (𝐶ℎ 𝑓 − 𝐶ℎ 𝑔) ≤ 2𝑘(𝑟) (ℎ𝛼 (𝑓) + ℎ𝛼 (𝑔)) ‖𝑓 − 𝑔‖𝐶 + 𝑘(𝑟)ℎ 𝛼 (𝑓 − 𝑔) ̂ ≤ 𝑘(𝑟)‖𝑓 − 𝑔‖𝐿𝑖𝑝𝛼 , ̂ as in the proof of Theorem 5.49, which proves the “if” part of The­ with the same 𝑘(𝑟) orem 5.51. To prove the “only if” part, let us now suppose that 𝐶ℎ satisfies a local Lipschitz condition (5.76) in the norm (0.71) of the space 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]). Putting again 𝑓(𝑡) ≡ 𝑢 and 𝑔(𝑡) ≡ 𝑣 in (5.76), we see that ℎ satisfies the local Lipschitz condition (5.13); we claim that ℎ󸀠 satisfies (5.14) for each 𝑟 > 0. Fix 𝑟 > 0 and choose 𝑓 ∈ 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) and 𝛿 > 0 such that ‖𝑓‖𝐿𝑖𝑝𝛼 < 𝑟 and ‖𝑓𝛿 ‖𝐿𝑖𝑝𝛼 ≤ 𝑟, where 𝑓𝛿 (𝑡) := 𝑓(𝑡) + 𝛿. By assumption, we then have ‖𝐶ℎ 𝑓𝛿 − 𝐶ℎ 𝑓‖𝐿𝑖𝑝𝛼 ≤ 𝐾(𝑟)‖𝑓𝛿 − 𝑓‖𝐿𝑖𝑝𝛼 = 𝐾(𝑟)𝛿 , and hence

|ℎ(𝑓(𝑡) + 𝛿) − ℎ(𝑓(𝑡))| ‖𝐶ℎ 𝑓𝛿 − 𝐶ℎ 𝑓‖𝐶 ≤ ≤ 𝐾(𝑟) 𝛿 𝛿 for all 𝑡 ∈ [𝑎, 𝑏] as well as |ℎ(𝑓(𝑠) + 𝛿) − ℎ(𝑓(𝑠)) − ℎ(𝑓(𝑡) + 𝛿) + ℎ(𝑓(𝑡))| ℎ𝛼 (𝐶ℎ 𝑓𝛿 − 𝐶ℎ 𝑓) ≤ ≤ 𝐾(𝑟) 𝛿|𝑠 − 𝑡|𝛼 𝛿

(5.84)

(5.85)

for all 𝑠, 𝑡 ∈ [𝑎, 𝑏]. Letting 𝛿 → 0 in (5.84) and (5.85), we conclude that |ℎ󸀠 (𝑓(𝑡))| ≤ 𝐾(𝑟),

|ℎ󸀠 (𝑓(𝑠)) − ℎ󸀠 (𝑓(𝑡))| ≤ 𝐾(𝑟) . |𝑠 − 𝑡|𝛼

This shows that the composition operator 𝐶ℎ󸀠 generated by ℎ󸀠 maps the space 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) into itself, by Theorem 5.24, and so the assertion follows. In the following Tables 5.14–5.16 we summarize our results on the “interaction” be­ tween properties of the function ℎ : ℝ → ℝ and those of the corresponding compo­ sition operator 𝐶ℎ : 𝑋 → 𝑋. We do this for the three spaces 𝑋 = 𝐶, 𝑋 = 𝐿𝑖𝑝, and 𝑋 = 𝐵𝑉 which have been the main focus in this chapter. Similar results hold for the more general spaces 𝑋 = 𝐿𝑖𝑝𝛼 (0 < 𝛼 < 1) and 𝑋 = 𝑊𝐵𝑉𝑝 (1 < 𝑝 < ∞). The hypotheses on ℎ and 𝐶ℎ get more and more restrictive in each row, i.e. the first row contains the mildest requirement, and the last row the strongest, at least formally.

374 | 5 Nonlinear composition operators Table 5.14. The operator 𝐶ℎ in 𝐶([𝑎, 𝑏]). ℎ ∈ 𝐶(ℝ) ℎ ∈ 𝐶(ℝ) ℎ ∈ 𝐶(ℝ) ℎ ∈ 𝐶(ℝ) ℎ ∈ 𝐿𝑖𝑝loc (ℝ) ℎ ∈ 𝐿𝑖𝑝(ℝ)

⇔ ⇔ ⇔ ⇔ ⇔ ⇔

𝐶ℎ 𝐶ℎ 𝐶ℎ 𝐶ℎ 𝐶ℎ 𝐶ℎ

: 𝐶 → 𝐶 bounded : 𝐶 → 𝐶 continuous : 𝐶 → 𝐶 uniformly continuous on bounded sets : 𝐶 → 𝐶 uniformly continuous on 𝐶 : 𝐶 → 𝐶 locally Lipschitz continuous : 𝐶 → 𝐶 globally Lipschitz continuous

Table 5.15. The operator 𝐶ℎ in 𝐿𝑖𝑝([𝑎, 𝑏]). ℎ ∈ 𝐿𝑖𝑝loc (ℝ) ℎ ∈ 𝐶1 (ℝ) ℎ ∈ 𝐶1 (ℝ) ℎ affine ℎ ∈ 𝐿𝑖𝑝1loc (ℝ) ℎ affine

⇔ ⇔ ⇔ ⇔ ⇔ ⇔

𝐶ℎ 𝐶ℎ 𝐶ℎ 𝐶ℎ 𝐶ℎ 𝐶ℎ

: 𝐿𝑖𝑝 → 𝐿𝑖𝑝 bounded : 𝐿𝑖𝑝 → 𝐿𝑖𝑝 continuous : 𝐿𝑖𝑝 → 𝐿𝑖𝑝 uniformly continuous on bounded sets : 𝐿𝑖𝑝 → 𝐿𝑖𝑝 uniformly continuous on 𝐿𝑖𝑝 : 𝐿𝑖𝑝 → 𝐿𝑖𝑝 locally Lipschitz continuous : 𝐿𝑖𝑝 → 𝐿𝑖𝑝 globally Lipschitz continuous

Table 5.16. The operator 𝐶ℎ in 𝐵𝑉([𝑎, 𝑏]). ℎ ∈ 𝐿𝑖𝑝loc (ℝ) ℎ ∈ 𝐶1 (ℝ) ℎ ∈ 𝐶1 (ℝ) ℎ affine ℎ ∈ 𝐿𝑖𝑝1loc (ℝ) ℎ affine

⇔ ⇒ ⇒ ⇒ ⇔ ⇔

𝐶ℎ 𝐶ℎ 𝐶ℎ 𝐶ℎ 𝐶ℎ 𝐶ℎ

: 𝐵𝑉 → 𝐵𝑉 bounded : 𝐵𝑉 → 𝐵𝑉 continuous : 𝐵𝑉 → 𝐵𝑉 uniformly continuous on bounded sets : 𝐵𝑉 → 𝐵𝑉 uniformly continuous on 𝐵𝑉 : 𝐵𝑉 → 𝐵𝑉 locally Lipschitz continuous : 𝐵𝑉 → 𝐵𝑉 globally Lipschitz continuous

We make some comments on these tables. Although the first four conditions on the right-hand side of Table 5.14 are formally independent of each other, the left-hand side show that they are actually all equivalent, and they all simply follow from the inclusion 𝐶ℎ (𝐶) ⊆ 𝐶. Moreover, global respectively local Lipschitz continuity of 𝐶ℎ in 𝐶([𝑎, 𝑏]) is reflected by exactly the same property of the underlying function ℎ, so no degeneracy for ℎ occurs. In Table 5.15, the situation is completely different. Only continuity and uniform continuity of 𝐶ℎ on bounded subsets of 𝐿𝑖𝑝([𝑎, 𝑏]) are equivalent, as we have proved in Theorem 5.26. Boundedness of 𝐶ℎ simply follows from the inclusion 𝐶ℎ (𝐿𝑖𝑝) ⊆ 𝐿𝑖𝑝. Moreover, local Lipschitz continuity of 𝐶ℎ in 𝐿𝑖𝑝([𝑎, 𝑏]) is equivalent to local Lipschitz continuity of ℎ󸀠 on the real line, and therefore holds for a reasonably large variety of nonlinear functions ℎ. On the other hand, imposing a global Lipschitz condition for 𝐶ℎ leads to a strong degeneracy for ℎ. Finally, less is known in the situation described in Table 5.16. As in Table 5.15, boundedness of 𝐶ℎ simply follows from the inclusion 𝐶ℎ (𝐵𝑉) ⊆ 𝐵𝑉, local Lipschitz continuity of 𝐶ℎ in 𝐵𝑉([𝑎, 𝑏]) is equivalent to local Lipschitz continuity of ℎ󸀠 on the real line, and imposing a global Lipschitz condition for 𝐶ℎ leads to a strong degeneracy for

5.5 Local Lipschitz continuity

| 375

ℎ. Unfortunately, only sufficient conditions on ℎ are known which imply the continuity of 𝐶ℎ , in one or the other sense, in 𝐵𝑉([𝑎, 𝑏]). At this moment, it is time to take a deep breath and to summarize, even at the risk of being redundant, what we have learned in this chapter about the autonomous composition operator (5.1) in many function spaces. – The operator 𝐶ℎ maps the space 𝐶([𝑎, 𝑏]) into itself if and only if the function ℎ is continuous on ℝ. In this case, 𝐶ℎ is automatically bounded and continuous. Moreover, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is globally Lipschitz on ℝ, and locally Lipschitz continuous if and only if ℎ is locally Lipschitz on ℝ. – The operator 𝐶ℎ maps the space 𝐶1 ([𝑎, 𝑏]) into itself if and only if the function ℎ is continuously differentiable on ℝ. In this case, 𝐶ℎ is automatically bounded and continuous. Moreover, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is affine, and locally Lipschitz continuous if and only if ℎ󸀠 is locally Lipschitz on ℝ. – The operator 𝐶ℎ maps the space 𝐿𝑖𝑝([𝑎, 𝑏]) into itself if and only if the function ℎ is locally Lipschitz on ℝ. In this case, 𝐶ℎ is automatically bounded, but not neces­ sarily continuous (Example 5.25). Moreover, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is affine, and locally Lipschitz continuous if and only if ℎ󸀠 is locally Lipschitz on ℝ. – The operator 𝐶ℎ maps the space 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) (0 < 𝛼 < 1) into itself if and only if the function ℎ is locally Lipschitz on ℝ. In this case, 𝐶ℎ is automatically bounded, but not necessarily continuous. Moreover, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is affine, and locally Lipschitz continuous if and only if ℎ󸀠 is locally Lipschitz on ℝ. – The operator 𝐶ℎ maps the space 𝐿𝑖𝑝1 ([𝑎, 𝑏]) into itself if and only if the function ℎ󸀠 is locally Lipschitz on ℝ. In this case, 𝐶ℎ is automatically bounded, but not neces­ sarily continuous (Example 5.40). Moreover, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is affine, and locally Lipschitz continuous if and only if ℎ󸀠󸀠 is locally Lipschitz on ℝ. – The operator 𝐶ℎ maps the space 𝐴𝐶([𝑎, 𝑏]) into itself if and only if the function ℎ is locally Lipschitz on ℝ. In this case, 𝐶ℎ is automatically bounded. Moreover, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is affine, and locally Lipschitz continuous if and only if ℎ󸀠 is locally Lipschitz on ℝ. The problem of characterizing continuity of 𝐶ℎ is open. – The operator 𝐶ℎ maps the space 𝐴𝐶1 ([𝑎, 𝑏]) into itself if and only if the function ℎ󸀠 is absolutely continuous on ℝ. In this case, 𝐶ℎ is automatically bounded. More­ over, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is affine, and locally Lip­ schitz continuous if and only if ℎ󸀠󸀠 is locally Lipschitz on ℝ. The problem of char­ acterizing continuity of 𝐶ℎ is open. – The operator 𝐶ℎ maps the space 𝐵𝑉([𝑎, 𝑏]) into itself if and only if the function ℎ is locally Lipschitz on ℝ. In this case, 𝐶ℎ is automatically bounded. Moreover, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is affine, and locally Lipschitz

376 | 5 Nonlinear composition operators

–

–

–

–

–

–

continuous if and only if ℎ󸀠 is locally Lipschitz on ℝ. The problem of characterizing continuity of 𝐶ℎ is open. The operator 𝐶ℎ maps the space 𝐵𝑉1 ([𝑎, 𝑏]) into itself if and only if the function ℎ󸀠 has locally bounded variation on ℝ. In this case, 𝐶ℎ is automatically bounded. Moreover, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is affine, and locally Lipschitz continuous if and only if ℎ󸀠󸀠 is locally Lipschitz on ℝ. The problem of characterizing continuity of 𝐶ℎ is open. The operator 𝐶ℎ maps the space 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) (1 < 𝑝 < ∞) into itself if and only if the function ℎ is locally Lipschitz on ℝ. In this case, 𝐶ℎ is automatically bounded. Moreover, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is affine, and locally Lipschitz continuous if and only if ℎ󸀠 is locally Lipschitz on ℝ. The problem of characterizing continuity of 𝐶ℎ is open. The operator 𝐶ℎ maps the space 𝑊𝐵𝑉1𝑝 ([𝑎, 𝑏]) (1 < 𝑝 < ∞) into itself if and only if the function ℎ󸀠 has locally bounded 𝑝-variation in Wiener’s sense on ℝ. In this case, 𝐶ℎ is automatically bounded. Moreover, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is affine, and locally Lipschitz continuous if and only if ℎ󸀠󸀠 is lo­ cally Lipschitz on ℝ. The problem of characterizing continuity of 𝐶ℎ is solved in Theorem 5.56 below. The operator 𝐶ℎ maps the space 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) (1 < 𝑝 < ∞) into itself if and only if the function ℎ is locally Lipschitz on ℝ. In this case, 𝐶ℎ is automatically bounded. Moreover, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is affine, and locally Lipschitz continuous if and only if ℎ󸀠 is locally Lipschitz on ℝ. The problem of characterizing continuity of 𝐶ℎ is open. The operator 𝐶ℎ maps the space 𝑅𝐵𝑉1𝑝 ([𝑎, 𝑏]) (1 < 𝑝 < ∞) into itself if and only if the function ℎ󸀠 has locally bounded 𝑝-variation in Riesz’s sense on ℝ. In this case, 𝐶ℎ is automatically bounded. Moreover, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is affine, and locally Lipschitz continuous if and only if ℎ󸀠󸀠 is locally Lipschitz on ℝ. The problem of characterizing continuity of 𝐶ℎ is open. The operator 𝐶ℎ maps the space Λ𝐵𝑉([𝑎, 𝑏]) into itself if and only if the function ℎ is locally Lipschitz on ℝ. In this case, 𝐶ℎ is automatically bounded. Moreover, 𝐶ℎ is globally Lipschitz continuous if and only if ℎ is affine, and locally Lipschitz continuous if and only if ℎ󸀠 is locally Lipschitz on ℝ. The problem of characterizing continuity of 𝐶ℎ is open.

We point out that this summary mostly contains conditions which are both necessary and sufficient. For the nonautonomous superposition operator (5.2), we are going to do the same in the next chapter; as we will see, the result will be quite different.

5.6 Comments on Chapter 5

| 377

5.6 Comments on Chapter 5 Although some of the results proved here for the composition operator (5.1) will be repeated in a more general setting in the next chapter for the superposition operator (5.2), we will give references for the most important theorems. The composition operator problem (COP) was formulated in a slightly different notation in [55, 56]. As we have seen, the local Lipschitz condition (5.13) for ℎ is neces­ sary and sufficient for the inclusion 𝐶ℎ (𝑋) ⊆ 𝑋 in many function spaces 𝑋 arising in applications. However, there are some exceptions. On the one hand, for some spaces 𝑋, condition (5.13) is too strong; for example, 𝐶ℎ (𝐶) ⊆ 𝐶 if and only if ℎ is continuous, see Theorem 5.20.¹⁷ On the other hand, for some other spaces 𝑋, it may also happen that condition (5.13) is too weak; this refers to all spaces of type 𝑋1 which we consid­ ered in Section 5.3. Apart from these spaces, there are other “exotic” spaces with the same property. To illustrate this, we recall a surprising result on functions having a primitive; the proof can be found in the paper [21]. Theorem 5.52. Let 𝐷−1 ([𝑎, 𝑏]) denote the space of all functions 𝑓 : [𝑎, 𝑏] → ℝ with a primitive, i.e. 𝑓 = 𝐹󸀠 for some differentiable function 𝐹. Then the operator (5.1) maps the space 𝐷−1 ([𝑎, 𝑏]) into itself if and only if the corresponding function ℎ is affine, i.e. has the form (5.68). As mentioned before, Theorem 5.9 due to Josephy [155] is the first example of a function space 𝑋 satisfying 𝐶𝑂𝑃(𝑋) = 𝐿𝑖𝑝loc (ℝ), namely, 𝑋 = 𝐵𝑉. The analogous result for 𝑋 = 𝐴𝐶 has been given in [219], for 𝑋 = 𝑅𝐵𝑉𝑝 in [221], and for 𝑋 = 𝐿𝑖𝑝 and 𝑋 = 𝐿𝑖𝑝𝛼 in [293–295].¹⁸ Of course, Theorem 5.10 which has been proved in [15] provides a unified approach to all of these results. The similar Theorem 5.12 is also taken from [14]. In [227], it is shown that 𝐶ℎ (𝑅𝐵𝑉𝑝 ) ⊆ 𝐵𝑉 if and only if ℎ ∈ 𝐿𝑖𝑝loc (ℝ); of course, this result follows from our Theorem 5.10. On the other hand, the requirement 𝐶ℎ (𝐵𝑉) ⊆ 𝐶 is so restrictive that it is fulfilled only for constant functions ℎ. To see this, suppose that ℎ(𝑢) ≠ ℎ(𝑣) for some 𝑢, 𝑣 ∈ ℝ. For fixed 𝑐 ∈ (𝑎, 𝑏), the function {𝑢 𝑓(𝑥) := 𝑢𝜒[𝑎,𝑐] (𝑥) + 𝑣𝜒(𝑐,𝑏] (𝑥) = { 𝑣 {

for 𝑎 ≤ 𝑥 ≤ 𝑐 , for 𝑐 < 𝑥 ≤ 𝑏

then belongs to 𝐵𝑉([𝑎, 𝑏]), but the function {ℎ(𝑢) 𝐶ℎ 𝑓(𝑥) := { ℎ(𝑣) {

for 𝑎 ≤ 𝑥 ≤ 𝑐 , for 𝑐 < 𝑥 ≤ 𝑏

17 Another example is given by Proposition 4.16 (a) which shows that 𝐶 ⊆ 𝐶𝑂𝑃(𝑅𝑆𝛼 ). 18 Unfortunately, the proof in these papers is false; a correct proof has been given in the recent survey paper [14].

378 | 5 Nonlinear composition operators is discontinuous at 𝑐. This implies, in particular, the result from [227] that 𝐶ℎ (𝐵𝑉) ⊆ 𝑅𝐵𝑉𝑝 (for some 𝑝 > 1) if and only if ℎ is constant. Theorem 5.13 may be found in [221], Theorems 5.14 and 5.15 in [251]. Table 5.1 is contained together with a detailed discussion of related results in the “folkloristic” paper [11]. There are many other function spaces 𝑋 with the characteristic property 𝐶𝑂𝑃(𝑋) = 𝐿𝑖𝑝loc (ℝ). For instance, Ul’yanov [306–308] has shown that this holds for the space 𝑋 = 𝐴 𝜔 ([0, 2𝜋) of all integrable functions whose Fourier–Haar coefficients satisfy a summability condition with weight 𝜔. The validity of the chain rule (5.38) and the change of variable formula (5.42) for absolutely continuous functions are dealt with of course in almost every textbook on real analysis and Lebesgue integration. Our discussion in Section 5.2 may be summarized as follows: in many function spaces 𝑋, the mere inclusion 𝐶ℎ (𝑋) ⊆ 𝑋 implies the boundedness of 𝐶ℎ in the under­ lying norm, while continuity of 𝐶ℎ in the underlying norm is a very delicate (and often open) question. As we pointed out before, three cases typically arise here: either we get continuity of 𝐶ℎ , in the same manner as boundedness, “for free,” or one knows precisely what additional condition on ℎ has to be imposed to get continuity of 𝐶ℎ , or the answer is simply unknown. The spaces 𝐶 and 𝐶1 fall into the first category, the spaces 𝐿𝑖𝑝 and 𝐿𝑖𝑝𝛼 into the second, and the spaces 𝐵𝑉 and 𝐴𝐶 (and many more) into the third. A simple sufficient condition for the continuity of 𝐶ℎ in the space 𝐵𝑉 which covers sufficiently many examples can be given; we state this in the following Proposition 5.53. Suppose that ℎ : ℝ → ℝ is real analytic. Then the corresponding operator (5.1) maps the space 𝐵𝑉([𝑎, 𝑏]) into itself and is both bounded and continuous in the norm (1.16). Proof. The assumption means that ℎ admits an expansion into a power series ∞

ℎ(𝑢) = ∑ 𝑎𝑘 𝑢𝑘 𝑘=0

with infinite radius of convergence. Since ℎ is then locally Lipschitz on ℝ, the operator 𝐶ℎ maps 𝐵𝑉([𝑎, 𝑏]) into itself and is bounded, by Theorem 5.9. It remains to show that 𝐶ℎ is continuous in the norm (1.16). Clearly, the composition operator 𝐶ℎ𝑛 generated by the 𝑛-th truncation 𝑛

ℎ𝑛 (𝑢) = ∑ 𝑎𝑘 𝑢𝑘 𝑘=0

of ℎ maps 𝐵𝑉([𝑎, 𝑏]) into itself since (𝐵𝑉([𝑎, 𝑏]), ‖ ⋅ ‖𝐵𝑉 ) is an algebra. Moreover, the difference ℎ − ℎ𝑛 is locally Lipschitz because sup |ℎ(𝑢) − ℎ𝑛 (𝑢) − ℎ(𝑣) + ℎ𝑛 (𝑣)| ≤ 𝑘1 (𝑟)|𝑢 − 𝑣| (𝑟 > 0)

|𝑢|≤𝑟

5.6 Comments on Chapter 5 |

379

with 𝑘1 (𝑟) := sup |ℎ󸀠 (𝑤) − ℎ󸀠𝑛(𝑤)| |𝑤|≤𝑟

(𝑟 > 0) .

This implies that ‖𝐶ℎ 𝑓 − 𝐶ℎ𝑛 𝑓‖𝐵𝑉 = |ℎ(𝑓(𝑎)) − ℎ𝑛 (𝑓(𝑎))| + Var(𝐶ℎ 𝑓 − 𝐶ℎ𝑛 𝑓; [𝑎, 𝑏]) ≤ |ℎ(𝑓(𝑎)) − ℎ𝑛(𝑓(𝑎))| + 𝑘1 (‖𝑓‖∞ ) Var(𝑓; [𝑎, 𝑏]) . However, this shows that the sequence (𝐶ℎ𝑛 𝑓)𝑛 converges to 𝐶ℎ 𝑓 in the norm (1.16) uniformly on each bounded subset of 𝐵𝑉([𝑎, 𝑏]). Consequently, the operator 𝐶ℎ is con­ tinuous in the norm (1.16) as well. In Section 5.3, we have seen that the composition operator problem has a completely different solution in spaces of differentiable functions like 𝐶1 , 𝐿𝑖𝑝1 , 𝐵𝑉1 , or 𝐴𝐶1 . Here, several new phenomena occur. First, a necessary and sufficient condition for the in­ clusion 𝐶ℎ (𝑋1 ) ⊆ 𝑋1 is often that ℎ belongs locally to the same space 𝑋1 . Second, working in a space of type 𝑋1 , we must make extensive use of the chain rule (5.38), and here the appearance of the factor 𝑓󸀠 (𝑥) on the right-hand side of (5.38) has a cer­ tain “regularizing effect.” For instance, before Theorem 5.37, we have discussed the phenomenon that 𝑓 ∈ 𝐶∞ and 𝑔 ∈ 𝐴𝐶loc (ℝ) does not imply that 𝑔 ∘ 𝑓 ∈ 𝐴𝐶, but multi­ plying by 𝑓󸀠 has the effect that (𝑔 ∘ 𝑓)𝑓󸀠 ∈ 𝐴𝐶. This effect also occurs in other function spaces. For example, for 𝑓(𝑥) := 𝑥2 and 𝑔(𝑦) := 1/√|𝑦| (0 < 𝑦 ≤ 1), we have 𝑓 ∈ 𝐶∞ ([0, 1]), 𝑔 ∈ 𝐿 1 ([0, 1]), 𝑔 ∘ 𝑓 ∈ ̸ 𝐿 1 ([0, 1]), (𝑔 ∘ 𝑓)𝑓󸀠 ∈ 𝐿 1 ([0, 1]) . For a slight generalization of this, see Exercise 5.15. Let us go back for a moment to the COP which consists of finding a necessary and sufficient condition for a fixed outer function ℎ under which ℎ ∘ 𝑓 belongs to a given class of functions whenever 𝑓 belongs to the same class. Sometimes, the following dual problem is also of interest: determine a necessary and sufficient condition for a fixed inner function 𝑓 under which ℎ ∘ 𝑓 belongs to a given class of functions whenever ℎ belongs to the same class. For example, for the class 𝐴𝐶, the solution of this problem goes back to Fichtengol’ts [114] who proved the following Theorem 5.54. Given 𝑓 : [𝑎, 𝑏] → [𝑐, 𝑑], the following two conditions are equivalent. (a) For every ℎ ∈ 𝐴𝐶([𝑐, 𝑑]), we have ℎ ∘ 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]). (b) The function 𝑓 is absolutely continuous, and the set 𝑓−1 (𝑦) ∩ [𝑎, 𝑏] only consists of isolated points for each 𝑦 ∈ [𝑐, 𝑑]. As Example 5.8 shows, the last condition cannot be dropped: both functions 𝑓 and ℎ in Example 5.8 are absolutely continuous, but 𝑓−1 (0) has the accumulation point 0. Since Janusz Matkowski seems to have been the first author who discovered the degeneracy phenomenon described in Section 5.4, the term spaces with the Matkowski property (our Definition 5.42) was introduced in [226]. At the beginning of Section 5.4,

380 | 5 Nonlinear composition operators we have given a short list of such spaces. The general Theorems 5.45 and 5.47 which are taken from [15] (see also [56] and [226]) show that we may add the spaces 𝑅𝐵𝑉𝑛𝑝 , 𝐵𝑉𝑛 , 𝑊𝐵𝑉𝑝 , 𝐵𝑉, 𝐴𝐶, and Λ 𝑞 𝐵𝑉 to this list. We point out, however, that the situation for the spaces 𝐵𝑉 and 𝑊𝐵𝑉𝑝 becomes more complicated in case of the nonautonomous superposition operator (5.2), see Definition 6.20, Theorem 6.19, and Example 6.21 in the next chapter. The fact that the global Lipschitz condition (5.67) for the operator 𝐶ℎ leads to the very special form (5.68) of ℎ has unpleasant consequences for applications: it means that whenever we want to apply Banach’s contraction mapping principle to a problem involving composition operators in a space with the Matkowski property, we can do this only if the problem is actually linear, and therefore not very interesting. This is the reason why one has to replace the global condition (5.67) by the local condition (5.76), as we have done in Section 5.5. Given a function 𝑓 : [𝑎, 𝑏] → ℝ and a partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } of [𝑎, 𝑏], recall that the second variation Var𝑊 2,1 (𝑓, 𝑃; [𝑎, 𝑏]) of 𝑓 on [𝑎, 𝑏] with respect to 𝑃 is defined by 󸀠󸀠 (2.141). In (2.150), we have established the important equality Var𝑊 2,1 (𝑓; [𝑎, 𝑏]) = ‖𝑓 ‖𝐿 1 for 𝑓 ∈ 𝐴𝐶1 ([𝑎, 𝑏]) which has been proven by Russell and may be viewed as a “second order analogue” to the classical formula (3.33). One could ask whether or not the re­ quirement that 𝑓 ∈ 𝐴𝐶1 ([𝑎, 𝑏]) is essential in Russell’s result. The following example shows that, in fact, functions exist with bounded second variation whose derivatives are not absolutely continuous. Example 5.55. Let 𝜑 : [0, 1] → ℝ be the Cantor function (3.6), and let 𝑓 : [0, 1] → ℝ be defined as in (5.60). We have already seen in Example 5.31 that 𝑓 ∈ ̸ 𝐴𝐶1 ([0, 1]). On the other hand, since 𝜑 is monotonically increasing, 𝑓 is convex, and thus belongs to ♥ 𝑊𝐵𝑉2,1 ([0, 1]). Example 5.55 shows that we cannot use Theorem 5.37 to solve the COP for the space 𝑊𝐵𝑉2,1 . As far as we know, the problem of describing 𝐶𝑂𝑃(𝑊𝐵𝑉2,1 ) is still open. As we have seen, the problem of characterizing those functions ℎ for which the corresponding operator 𝐶ℎ is continuous in norm is open for many spaces. We state this for absolutely continuous functions as Problem 5.1. Suppose that ℎ : ℝ → ℝ satisfies condition (5.13). Does this imply that the corresponding operator 𝐶ℎ is continuous in the norm (3.42) or the norm (3.43) of the space 𝐴𝐶([𝑎, 𝑏])? Problem 5.2. Same question as in Problem 5.1 for the spaces 𝐵𝑉([𝑎, 𝑏]) and 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) (𝑝 > 1). Surprisingly enough, in [56], this problem was solved for the space 𝑊𝐵𝑉1𝑝 in case 𝑝 > 1, i.e. for primitives of 𝑊𝐵𝑉𝑝 -functions. More precisely, the authors of [56] prove the fol­ lowing

5.6 Comments on Chapter 5 |

381

Theorem 5.56. For 1 < 𝑝 < ∞, the operator (5.1) maps the space 𝑊𝐵𝑉1𝑝 ([𝑎, 𝑏]) into itself if and only if ℎ ∈ 𝑊𝐵𝑉1𝑝,𝑙𝑜𝑐 (ℝ). In this case, the operator (5.1) is automatically bounded in the norm (5.54). Moreover, the operator (5.1) is continuous in the norm (5.54) if and only if ℎ belongs to the closure of 𝑊𝐵𝑉1𝑝,𝑙𝑜𝑐 (ℝ) ∩ 𝐶∞ (ℝ) in the norm (5.54). Of course, the closure of 𝑊𝐵𝑉1𝑝,𝑙𝑜𝑐 (ℝ) ∩ 𝐶∞ (ℝ) in the norm (5.54) is a proper sub­ space of 𝑊𝐵𝑉1𝑝,𝑙𝑜𝑐 (ℝ). We believe that a similar restriction is needed in the Riesz space 𝑅𝐵𝑉1𝑝 ([𝑎, 𝑏]), but we were unable to prove it. Problem 5.3. Find a condition on ℎ, both necessary and sufficient, under which the cor­ responding operator (5.1) is continuous in the space 𝑅𝐵𝑉1𝑝 ([𝑎, 𝑏]) with norm (5.55). Problem 5.4. Find a condition on ℎ, both necessary and sufficient, under which the cor­ responding operator (5.1) maps the space 𝑊𝐵𝑉2,1 ([𝑎, 𝑏]) into itself. The COP for the space 𝜅𝐵𝑉 of functions of bounded Korenblum variation was solved recently in the paper [17], where it is shown that 𝐶ℎ (𝜅𝐵𝑉) ⊆ 𝜅𝐵𝑉 if and only if ℎ ∈ 𝐿𝑖𝑝loc (ℝ). Moreover, in this case, the operator (5.1) is automatically bounded in the norm (2.136). Most of the material covered in Section 5.5 has been taken from the paper [18]. Locally Lipschitz composition operators in the Schramm space 𝛷𝐵𝑉 (which is missing in the summary at the end of Section 5.5) are discussed in [230]. In this chapter, we have only considered functions over some compact interval. Superpositions of functions of several variables have been studied by Chistyakov in a series of papers [85–95]. Since these results refer to the nonautonomous case of the operator (5.2), we postpone them to the next chapter. However, following [191], we briefly consider a special result on composition oper­ ators which are generated by an outer function ℎ of two variables. In the legendary Scot­ tish Book, the mathematician Max Eidelheit posed the following problem for functions which are absolutely continuous “on traces:” given a function ℎ : [0, 1] × [0, 1] → ℝ, suppose that ℎ(𝑢, ⋅) ∈ 𝐴𝐶([0, 1]) for every 𝑢 ∈ [0, 1], and ℎ(⋅, 𝑣) ∈ 𝐴𝐶([0, 1]) for every 𝑣 ∈ [0, 1], and let 𝑓 : [0, 1] → [0, 1] and 𝑔 : [0, 1] → [0, 1] also be absolutely continuous functions. Does it follow that the function 𝑥 󳨃→ ℎ(𝑓(𝑥), 𝑔(𝑥)) is absolutely continuous on [0, 1]? If not, then perhaps this is true under the additional assumption that 1 1

1 1 𝑝

∫ ∫ |ℎ𝑢 (𝑢, 𝑣)| 𝑑𝑣 𝑑𝑢 < ∞,

∫ ∫ |ℎ𝑣 (𝑢, 𝑣)|𝑝 𝑑𝑣 𝑑𝑢 < ∞

0 0

0 0

(5.86)

for some 𝑝 > 1? The next example shows that the answer to Eidelheit’s first question is negative. Example 5.57. Define ℎ : [0, 1] × [0, 1] → ℝ by { 2𝑢𝑣 2 2 ℎ(𝑢, 𝑣) := { 𝑢 +𝑣 0 {

for (𝑢, 𝑣) ≠ (0, 0) , for (𝑢, 𝑣) = (0, 0) ,

(5.87)

382 | 5 Nonlinear composition operators and let 𝑓(𝑥) = 𝑔(𝑥) := 𝑥. Then |ℎ(𝑢, 𝑣) − ℎ(𝑢󸀠 , 𝑣)| ≤ 𝑣2 |𝑢 − 𝑢󸀠 | for 0 < 𝑣 ≤ 1, and similarly |ℎ(𝑢, 𝑣) − ℎ(𝑢, 𝑣󸀠 )| ≤ 𝑢2 |𝑣 − 𝑣󸀠 | for 0 < 𝑢 ≤ 1; moreover, ℎ(𝑢, 0) = ℎ(0, 𝑣) ≡ 0. Consequently, all hypotheses of Eidelheit’s problem are satisfied, but the function ℎ(𝑓(𝑥), 𝑔(𝑥)) = ℎ(𝑥, 𝑥) = 𝜒(0,1] is not continuous, let alone absolutely continuous. ♥ Note that the integrability condition (5.86) fails for the function (5.87) for every 𝑝 > 1 (Exercise 5.27), and so this function does not provide a counterexample to Eidelheit’s second question. However, one may show that Eidelheit’s second question also has a negative answer. In fact, consider the function ℎ : [0, 1] × [0, 1] → ℝ defined by 𝑢

ℎ(𝑢, 𝑣) := ∫ log 𝑡 𝑑𝑡

(0 ≤ 𝑢 ≤ 1)

(5.88)

0

which depends on 𝑣 only formally. Then 1 1

1 1

1 𝑝

𝑝

∫ ∫ |ℎ𝑢 (𝑢, 𝑣)| 𝑑𝑣 𝑑𝑢 = ∫ | log 𝑡| 𝑑𝑡 < ∞,

∫ ∫ |ℎ𝑣 (𝑢, 𝑣)|𝑝 𝑑𝑣 𝑑𝑢 = 0

0 0

0 0

0

for all 𝑝 > 1, but¹⁹ ℎ𝑢 (⋅, 𝑣) ∈ ̸ 𝐿 ∞ ([0, 1]). Now, the trick consists of choosing 𝑓 = 𝑔 ∈ 𝐴𝐶([0, 1]) in such a way²⁰ that the map 𝑥 󳨃→ ℎ(𝑓(𝑥), 𝑔(𝑥)) is not absolutely continuous on [0, 1]. From Theorem 5.10, we know that the outer function ℎ must not be locally Lipschitz in this case; but this is true by Theorem 3.20.

5.7 Exercises to Chapter 5 We state some exercises on the topics covered in this chapter; exercises marked with an asterisk * are more difficult. Exercise 5.1. Let 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) and let ℓ : [𝑐, 𝑑] → [𝑎, 𝑏] be increasing and absolutely continuous with ℓ(𝑐) = 𝑎 and ℓ(𝑑) = 𝑏. Show that 𝑓 ∘ ℓ ∈ 𝐴𝐶([𝑐, 𝑑]). Is the same true if ℓ is only increasing and continuous? Exercise 5.2. Let 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) and ℎ ∈ 𝐴𝐶loc (ℝ). Prove that ℎ ∘ 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) if and only if ℎ ∘ 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]). Compare with Example 5.8. Exercise 5.3. Let 𝑓 : [𝑎, 𝑏] → [𝑐, 𝑑] be monotone and ℎ ∈ 𝐵𝑉([𝑐, 𝑑]). Show that ℎ ∘ 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏]) and compare with Example 5.8.

19 We have already seen in the similar Example 0.12 that log ∈ 𝐿 𝑝 ([0, 1]) for every 𝑝 ≥ 1, but log ∉ 𝐿 ∞ ([0, 1]). 20 In case of the function ℎ(𝑢) = √𝑢, we have achieved this by choosing 𝑓 as in (5.11); for the function (5.88), this has to be modified accordingly.

5.7 Exercises to Chapter 5 | 383

Exercise 5.4. Prove or disprove the following converse of Lemma 5.2: every COP-in­ variant function space 𝑋 is shift-invariant. Exercise 5.5. Is the space 𝛷𝐵𝑉([𝑎, 𝑏]) COP-invariant for all Schramm sequences 𝛷? If not, find additional conditions on 𝛷 under which this is true. Exercise 5.6*. Formulate and prove an analogous result to Theorem 5.14 for the space 𝛷𝐵𝑉([𝑎, 𝑏]), where 𝛷 is an arbitrary Schramm sequence. Exercise 5.7. Solve again Exercise 3.5 by means of Theorem 5.10 and Example 5.8. Exercise 5.8. Let 𝜙 be a Young function which satisfies the 𝛿2 -condition (2.4), but not the 01 condition (2.17). Suppose that the operator (5.1) maps the space 𝜙𝐵𝑉([𝑎, 𝑏]) into itself. Imitating the proof of Proposition 5.7, show that the generating function ℎ is then continuous. Exercise 5.9. Is the hypothesis ℎ ∈ 𝐶1 (ℝ) sufficient for the corresponding operator 𝐶ℎ to be continuous in the norm (3.42) of the space 𝐴𝐶([𝑎, 𝑏])? Exercise 5.10. Is the hypothesis ℎ ∈ 𝐶1 (ℝ) sufficient for the corresponding operator 𝐶ℎ to be continuous in the norm (2.90) of the space 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏])? Exercise 5.11. Suppose that ℎ : ℝ → ℝ has the property that the corresponding com­ position operator (5.1) maps a function space 𝑋 into itself and is bounded in the norm of 𝑋. For 𝑟 > 0, we put 𝜇𝑟 (ℎ, 𝑋) := sup {‖𝐶ℎ 𝑓‖𝑋 : ‖𝑓‖𝑋 ≤ 𝑟} = sup {‖ℎ ∘ 𝑓‖𝑋 : ‖𝑓‖𝑋 ≤ 𝑟} and call this characteristic the growth function of ℎ in 𝑋. Calculate the functions 𝜇𝑟 (ℎ, 𝐶) and 𝜇𝑟 (ℎ, 𝐶1 ) under the hypotheses of Theorem 5.20 and Theorem 5.30, re­ spectively. Exercise 5.12. Using the notation of Theorem 5.24, give an upper estimate for the growth function 𝜇𝑟 (ℎ, 𝐿𝑖𝑝𝛼 ) from Exercise 5.11 in the space 𝐿𝑖𝑝𝛼 ([0, 1]). Exercise 5.13. Let 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) with 𝑓([𝑎, 𝑏]) ⊆ [𝑐, 𝑑], and let 𝑔 ∈ 𝐿 1 ([𝑐, 𝑑]). If (𝑔 ∘ 𝑓)𝑓󸀠 ∈ 𝐿 1 ([𝑎, 𝑏]), then 𝑓(𝛽)

𝛽

∫ 𝑔(𝑦) 𝑑𝑦 = ∫ 𝑔(𝑓(𝑡))𝑓󸀠 (𝑡) 𝑑𝑡 𝑓(𝛼)

𝛼

for every interval [𝛼, 𝛽] ⊂ [𝑎, 𝑏]. Compare with Theorem 5.18. Exercise 5.14. In the notation of Exercise 5.13, prove that (𝑔 ∘ 𝑓)𝑓󸀠 ∈ 𝐿 1 ([𝑎, 𝑏]) if 𝑓 ∈ 𝐴𝐶([𝑎, 𝑏]) is monotone with 𝑓([𝑎, 𝑏]) ⊆ [𝑐, 𝑑] and 𝑔 ∈ 𝐿 1 ([𝑐, 𝑑]). Exercise 5.15. For 𝛼 ∈ ℝ+ and 𝛽 ∈ ℝ, define 𝑓 : [0, 1] → [0, 1] by 𝑓(𝑥) := 𝑥𝛼 and 𝑔 : (0, 1] → ℝ by 𝑔(𝑦) := 𝑦𝛽 . Determine all pairs (𝛼, 𝛽) for which 𝑔 ∘ 𝑓 ∈ 𝐿 1 ([0, 1]) and all pairs (𝛼, 𝛽) for which (𝑔 ∘ 𝑓)𝑓󸀠 ∈ 𝐿 1 ([0, 1]).

384 | 5 Nonlinear composition operators Exercise 5.16. Suppose that 𝑓 ∈ 𝐶∞ ([𝑎, 𝑏]) and 𝑔 ∈ 𝐵𝑉loc (ℝ). Does it follow that (𝑔 ∘ 𝑓)𝑓󸀠 ∈ 𝐵𝑉([𝑎, 𝑏)? Exercise 5.17. Using the notation of Theorem 5.35, give an upper estimate for the growth function 𝜇𝑟 (ℎ, 𝐵𝑉1 ) from Exercise 5.11 in the space 𝐵𝑉1 ([0, 1]). Exercise 5.18. Using the notation of Theorem 5.37, give an upper estimate for the growth function 𝜇𝑟 (ℎ, 𝐴𝐶1 ) from Exercise 5.11 in the space 𝐴𝐶1 ([0, 1]). Exercise 5.19. Using the notation of Theorem 5.38, give an upper estimate for the growth function 𝜇𝑟 (ℎ, 𝑅𝐵𝑉1𝑝 ) from Exercise 5.11 in the space 𝑅𝐵𝑉1𝑝 ([0, 1]). Exercise 5.20. Using the notation of Theorem 5.39, give an upper estimate for the growth function 𝜇𝑟 (ℎ, 𝐿𝑖𝑝1 ) from Exercise 5.11 in the space 𝐿𝑖𝑝1 ([0, 1]). Exercise 5.21. Is the hypothesis ℎ ∈ 𝐶2 (ℝ) sufficient for the corresponding operator 𝐶ℎ to be continuous in the norm (5.53) of the space 𝐴𝐶1 ([𝑎, 𝑏])? Exercise 5.22. Is the hypothesis ℎ ∈ 𝐶2 (ℝ) sufficient for the corresponding operator 𝐶ℎ to be continuous in the norm (5.54) of the space 𝑊𝐵𝑉1𝑝 ([𝑎, 𝑏])? Exercise 5.23. Is the hypothesis ℎ ∈ 𝐶2 (ℝ) sufficient for the corresponding operator 𝐶ℎ to be continuous in the norm (5.55) of the space 𝑅𝐵𝑉1𝑝 ([𝑎, 𝑏])? Exercise 5.24. Imitate the proof of Theorem 5.26 to prove Theorem 5.41. Exercise 5.25. Suppose that the composition operator 𝐶ℎ generated by some function ℎ : ℝ → ℝ maps the space 𝐶1 ([𝑎, 𝑏]) into itself and is continuous and bounded with respect to one of the norms in (0.65). Moreover, assume that ℎ󸀠 exists on ℝ and satisfies (5.14). Show that the operator 𝐶ℎ generated by ℎ satisfies then (5.76). Exercise 5.26. Prove the following converse of Exercise 5.6. Suppose that the compo­ sition operator 𝐶ℎ generated by ℎ maps 𝐶1 ([𝑎, 𝑏]) into itself and satisfies (5.76). Show that then ℎ󸀠 exists on ℝ and satisfies (5.14). Exercise 5.27. Let ℎ : [0, 1] × [0, 1] → ℝ be defined by (5.87). Show that 1 1

1 1 𝑝

∫ ∫ |ℎ𝑢 (𝑢, 𝑣)| 𝑑𝑣 𝑑𝑢 = ∫ ∫ |ℎ𝑣 (𝑢, 𝑣)|𝑝 𝑑𝑣 𝑑𝑢 = ∞ 0 0

0 0

for every 𝑝 > 1. Exercise 5.28. Deduce from the estimate (5.83) the degeneracy result contained in Theorem 5.47 for the space 𝐵𝑉 = 𝑊𝐵𝑉1 .

6 Nonlinear superposition operators When we replace the autonomous operator 𝐶ℎ 𝑓(𝑥) = ℎ(𝑓(𝑥)) by the nonautonomous operator 𝑆ℎ 𝑓(𝑥) = ℎ(𝑥, 𝑓(𝑥)), we encounter many new phenomena. For example, even necessary and sufficient criteria on ℎ under which the corresponding operator 𝑆ℎ maps a function space into itself are only known in quite exceptional cases. Moreover, while the operator 𝐶ℎ is often both automatically bounded and continuous in standard func­ tion spaces, it may happen that the operator 𝑆ℎ is bounded but discontinuous, or con­ tinuous but unbounded. Imposing a global Lipschitz condition on 𝑆ℎ in norm leads to the same degeneracy phenomena as for the operator 𝐶ℎ , which is not true for local Lipschitz conditions in norm. In this chapter, we will also study the uniform bound­ edness and uniform continuity of the operator 𝑆ℎ . As a rule, for all of these properties of 𝑆ℎ , one may only formulate sufficient conditions, while their necessity is often an open problem.

6.1 Boundedness and continuity Given a function ℎ : [𝑎, 𝑏] × ℝ → ℝ, we are now going to study the (nonautonomous) superposition operator 𝑆ℎ defined by 𝑆ℎ 𝑓(𝑥) = ℎ(𝑥, 𝑓(𝑥))

(𝑎 ≤ 𝑥 ≤ 𝑏) ,

(6.1)

where 𝑓 is taken from some space of real functions on [𝑎, 𝑏]. It turns out that the op­ erator (6.1) is far more complicated than its autonomous counterpart (5.1) which we studied in great detail in Chapter 5. The main reason for this is the “interaction” be­ tween the two variables of the function ℎ which, in some cases, may lead to a quite pathological behavior of the corresponding operator 𝑆ℎ . Fortunately, there are some exceptions from this rule: in a few simple function spaces, the operator (6.1) behaves in exactly the same way as the operator (5.1). For example, the following two results hold which are precise analogues to Theorems 5.20 and 5.21: Theorem 6.1. The operator (6.1) maps the space 𝐶([𝑎, 𝑏]) into itself if and only if the function ℎ is continuous on [𝑎, 𝑏] × ℝ. In this case, the operator (6.1) is automatically bounded and continuous in the norm (0.45). Theorem 6.2. For 1 ≤ 𝑝, 𝑞 < ∞, the superposition operator (6.1) maps the space 𝐿 𝑝 ([𝑎, 𝑏]) into the space 𝐿 𝑞 ([𝑎, 𝑏]) if and only if the function ℎ satisfies the growth con­ dition |ℎ(𝑥, 𝑢)| ≤ 𝛼(𝑥) + 𝛽|𝑢|𝑝/𝑞 (𝑎 ≤ 𝑥 ≤ 𝑏, 𝑢 ∈ ℝ) (6.2) for some function 𝛼 ∈ 𝐿 𝑞 ([𝑎, 𝑏]) and some constant 𝛽 > 0. In this case, the operator (6.1) is automatically bounded and continuous in the norm (0.11).

386 | 6 Nonlinear superposition operators The proof of Theorem 6.1 is a simple consequence of the Tietze–Urysohn extension lemma, while the proof of Theorem 6.2 relies on a classical result of Krasnosel’skij ([165–168] or [170]). Note that in case 𝑝 = 𝑞, condition (6.2) reads |ℎ(𝑥, 𝑢)| ≤ 𝛼(𝑥) + 𝛽|𝑢| (𝑎 ≤ 𝑥 ≤ 𝑏, 𝑢 ∈ ℝ),

(6.3)

involving a function 𝛼 ∈ 𝐿 𝑝 ([𝑎, 𝑏]) and a constant 𝛽 > 0; this means that 𝑆ℎ (𝐿 𝑝 ) ⊆ 𝐿 𝑝 if and only if the map 𝑢 󳨃→ ℎ(𝑥, 𝑢) has sublinear growth for large values of |𝑢|. The following results and counterexamples are in sharp contrast to Theorems 6.1 and 6.2. In fact, we take this opportunity to recall some pathological phenomena of the nonautonomous superposition operator (6.1) in other function spaces which are somewhat beyond the scope of this monograph. All of these phenomena are possible only due to a complicated “interaction” between the variables 𝑥 and 𝑢 in the function (𝑥, 𝑢) 󳨃→ ℎ(𝑥, 𝑢) which is “hidden” in the autonomous case of the composition operator (5.1). To begin with, consider for 0 < 𝛼 ≤ 1 the Banach space 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) of all Hölder continuous (in particular, Lipschitz continuous for 𝛼 = 1) functions on [𝑎, 𝑏], equipped with one of the equivalent norms (0.71) or (0.77). Interestingly, in this space, the su­ perposition operator (6.1) exhibits a rather pathological behavior: Example 6.3. Let ℎ : [0, 1] × ℝ → ℝ be defined by {0 ℎ(𝑥, 𝑢) := { 1 − { 𝑢2/𝛼

if 𝑢 ≤ 𝑥𝛼/2 , 𝑥 𝑢4/𝛼

if 𝑢 > 𝑥𝛼/2 .

(6.4)

A somewhat cumbersome calculation then shows that the operator (6.1) generated by this function maps 𝐿𝑖𝑝𝛼 ([0, 1]) into itself, but ℎ is discontinuous at (0, 0), and so 𝑆ℎ does not map 𝐶([0, 1]) into itself! ♥ We point out that the reason for the pathological behavior of the function ℎ in Exam­ ple 6.3 is the lack of boundedness of the corresponding superposition operator 𝑆ℎ in the norm (0.71). In fact, the sequence (𝑓𝑛 )𝑛 of constant functions 𝑓𝑛 (𝑥) ≡ 1/𝑛 satisfies ‖𝑓𝑛 ‖𝐿𝑖𝑝𝛼 =

1 , 𝑛

‖𝑆ℎ 𝑓𝑛 ‖𝐿𝑖𝑝𝛼 ≥ |𝑆ℎ 𝑓𝑛 (0)| = 𝑛2/𝛼 → ∞ (𝑛 → ∞) .

If we add boundedness to the acting condition 𝑆ℎ (𝐿𝑖𝑝𝛼 ) ⊆ 𝐿𝑖𝑝𝛼 , however, we get the result that we expect (see [20, Theorem 7.3] for the proof): Theorem 6.4. The operator (6.1) maps the space 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) into itself and is bounded with respect to the norm (0.71) if and only if the function ℎ satisfies the mixed local Hölder–Lipschitz condition |ℎ(𝑠, 𝑢) − ℎ(𝑡, 𝑣)| ≤ 𝑘(𝑟) (|𝑠 − 𝑡|𝛼 + |𝑢 − 𝑣|)

(𝑎 ≤ 𝑠, 𝑡 ≤ 𝑏; |𝑢|, |𝑣| ≤ 𝑟) .

In particular, the function ℎ is then necessarily continuous on [𝑎, 𝑏] × ℝ.

(6.5)

6.1 Boundedness and continuity

|

387

Of course, in the special case of the autonomous composition operator (5.1), condition (6.5) takes the simpler form |ℎ(𝑢) − ℎ(𝑣)| ≤ 𝑘(𝑟)|𝑢 − 𝑣| (|𝑢|, |𝑣| ≤ 𝑟)

(6.6)

which is precisely condition (5.13) and so reproduces Theorem 5.24. One could ask if it is possible to give a necessary and sufficient condition for the mere inclusion 𝑆ℎ (𝐿𝑖𝑝𝛼 ) ⊆ 𝐿𝑖𝑝𝛼 , i.e. without requiring boundedness of the operator 𝑆ℎ . Such condi­ tions are in fact known, but there are very clumsy and hard to verify in practice. For the sake of completeness, we cite a result whose (quite technical) proof can be found in [20, Theorem 7.1]. To this end, we denote for (𝑠0 , 𝑢0 ) ∈ [𝑎, 𝑏] × ℝ, 𝑟 > 0, 𝛿 > 0, and 𝛼 ∈ (0, 1], the “bow-tie” region 𝑊𝛼 (𝑠0 , 𝑢0 , 𝑟, 𝛿) := {(𝑠, 𝑢) ∈ [𝑎, 𝑏] × ℝ : |𝑠 − 𝑠0 | ≤ 𝛿, |𝑢 − 𝑢0 | ≤ 𝑟|𝑠 − 𝑠0 |𝛼 } . Proposition 6.5. The operator (6.1) maps the space 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) into itself if and only if for all (𝑠0 , 𝑢0 ) ∈ [𝑎, 𝑏] × ℝ and all 𝑟 > 0, one may find 𝛿 > 0 and 𝑘(𝑟) > 0 such that |ℎ(𝑠, 𝑢) − ℎ(𝑡, 𝑣)| ≤ 𝑘(𝑟) (|𝑠 − 𝑡|𝛼 +

|𝑢 − 𝑣| ) 𝑟

for all (𝑠, 𝑢), (𝑡, 𝑣) ∈ 𝑊𝛼 (𝑠0 , 𝑢0 , 𝑟, 𝛿). Now, we study the superposition operator (6.1) in the space 𝐶1 ([𝑎, 𝑏]) of continuously differentiable functions 𝑓 : [𝑎, 𝑏] → ℝ, equipped with the first norm in (0.65). Since we are going to deal with differentiable functions, we have to consider the derivative 𝑔󸀠 (𝑥) = 𝐷1 ℎ(𝑥, 𝑓(𝑥)) + 𝐷2 ℎ(𝑥, 𝑓(𝑥))𝑓󸀠 (𝑥)

(6.7)

of 𝑔 = 𝑆ℎ 𝑓, where 𝐷1 ℎ denotes the partial derivative of ℎ with respect to the first argu­ ment and 𝐷2 ℎ the partial derivative of ℎ with respect to the second argument. Surprisingly enough, in contrast to the space 𝐶([𝑎, 𝑏]), the operator 𝑆ℎ has a quite unexpected behavior in the space 𝐶1 ([𝑎, 𝑏]). For example, it may happen that 𝑆ℎ maps 𝐶1 into itself, although the generating function ℎ is discontinuous (and so 𝑆ℎ does not map 𝐶 into itself)! Since such pathologies seem to be interesting, we briefly recall an example due to Matkowski (see Section 8.2 in [20]). Example 6.6. Let ℎ : [0, 1] × ℝ → ℝ be defined by 0 { { { 𝑢2 𝑢3 𝑓(𝑥, 𝑢) := {3 𝑥 − 2 𝑥√𝑥 { { {1

if 𝑢 ≤ 0 , if 0 < 𝑢 < √𝑥

(6.8)

if 𝑢 ≥ √𝑥 .

As in Example 6.3, a rather cumbersome but straightforward calculation then shows that the operator (6.1) generated by this function maps 𝐶1 ([0, 1]) into itself, but ℎ is obviously discontinuous at (0, 0), and so 𝑆ℎ does not map 𝐶([0, 1]) into itself! ♥

388 | 6 Nonlinear superposition operators Theorem 6.4 shows that the reason of the pathological behavior of the function ℎ in Example 6.3 is the lack of boundedness of the corresponding superposition operator 𝑆ℎ in the norm (0.71). On the other hand, the reason for the pathological behavior of the function ℎ in Example 6.6 is the lack of continuity of the corresponding superposition operator 𝑆ℎ in the norm (0.65). In fact, considering again the sequence (𝑓𝑛)𝑛 of constant functions 𝑓𝑛 (𝑥) ≡ 1/𝑛, we obtain ‖𝑓𝑛 ‖𝐶1 =

1 → 0, 𝑛

‖𝑆ℎ 𝑓𝑛 ‖𝐶1 ≥ |𝑆ℎ 𝑓𝑛 (0)| = 1 ↛ 0

(𝑛 → ∞) .

If we add continuity to the acting condition 𝑆ℎ (𝐶1 ) ⊆ 𝐶1 , however, we get the result that we expect; this time, we give a complete proof. Theorem 6.7. The operator (6.1) maps the space 𝐶1 ([𝑎, 𝑏]) into itself and is continuous with respect to the norm (0.65) if and only if the function ℎ is continuously differentiable on [𝑎, 𝑏] × ℝ. Proof. The “if” part is an immediate consequence of the chain rule (6.7) and the defi­ nition of the norm (0.65) in the space 𝐶1 ([𝑎, 𝑏]); thus, we only have to prove the “only if” part. Suppose that 𝑆ℎ maps the space 𝐶1 ([𝑎, 𝑏]) into itself and is continuous with respect to the norm (0.65). We have to show that the partial derivatives 𝐷1 ℎ and 𝐷2 ℎ of ℎ are continuous functions. Writing 𝑓𝑢 for the constant function 𝑓𝑢 (𝑥) ≡ 𝑢 and 𝑔𝑢 = 𝑆ℎ 𝑓𝑢 , we have 𝑔𝑢 (𝑥) = ℎ(𝑥, 𝑢), and thus 𝑔𝑢󸀠 (𝑥) = 𝐷1 ℎ(𝑥, 𝑢). Given 𝜀 > 0, choose 𝛿 > 0 such that |𝑠 − 𝑡| ≤ 𝛿 and |𝑢 − 𝑣| = ‖𝑓𝑢 − 𝑓𝑣 ‖𝐶1 ≤ 𝛿 implies |𝑔𝑢󸀠 (𝑠) − 𝑔𝑢󸀠 (𝑡)| ≤ 𝜀,

‖𝑔𝑢 − 𝑔𝑣 ‖𝐶1 ≤ 𝜀 ,

which is possible by the uniform continuity of 𝑔𝑢󸀠 on [𝑎, 𝑏] and our continuity assump­ tion on the operator 𝑆ℎ . Then |𝐷1 ℎ(𝑠, 𝑢) − 𝐷1 ℎ(𝑡, 𝑣)| = |𝑔𝑢󸀠 (𝑠) − 𝑔𝑣󸀠 (𝑡)| ≤ |𝑔𝑢󸀠 (𝑠) − 𝑔𝑢󸀠 (𝑡)| + |𝑔𝑢󸀠 (𝑡) − 𝑔𝑣󸀠 (𝑡)| ≤ 𝜀 + ‖𝑔𝑢󸀠 − 𝑔𝑣󸀠 ‖𝐶 ≤ 𝜀 + ‖𝑔𝑢 − 𝑔𝑣 ‖𝐶1 ≤ 2𝜀 for |𝑠 − 𝑡| ≤ 𝛿 and |𝑢 − 𝑣| ≤ 𝛿, which shows that 𝐷1 ℎ is indeed continuous on [𝑎, 𝑏] × ℝ. The proof of the continuity of 𝐷2 ℎ on [𝑎, 𝑏] × ℝ is somewhat harder. Writing 𝑓𝑡,𝑣 for the linear function 𝑓𝑡,𝑣 (𝑥) = 𝑣 − 𝑡 + 𝑥 and 𝑔𝑡,𝑣 = 𝑆ℎ 𝑓𝑡,𝑣 , we have 𝑔𝑡,𝑣 (𝑥) = ℎ(𝑥, 𝑣 − 𝑡 + 𝑥), and hence 󸀠 𝑔𝑡,𝑣 (𝑥) = 𝐷1 ℎ(𝑥, 𝑣 − 𝑡 + 𝑥) + 𝐷2 ℎ(𝑥, 𝑣 − 𝑡 + 𝑥)

6.1 Boundedness and continuity

| 389

by (6.7). For fixed 𝑠0 ∈ [𝑎, 𝑏], 𝑢0 ∈ ℝ, and 𝜏 > 0, we have ℎ(𝑠0 , 𝑢0 + 𝜏) − ℎ(𝑠0 , 𝑢0 ) = ℎ(𝑠0 + 𝜏, 𝑢0 + 𝜏) − ℎ(𝑠0 , 𝑢0 ) − ℎ(𝑠0 + 𝜏, 𝑢0 + 𝜏) + ℎ(𝑠0 , 𝑢0 + 𝜏) 𝑠0 +𝜏

= 𝑔𝑠0 ,𝑢0 (𝑠0 + 𝜏) − 𝑔𝑠0 ,𝑢0 (𝑠0 ) − ∫ 𝐷1 ℎ(𝜎, 𝑢0 + 𝜏) 𝑑𝜎 𝑠0

= 𝑔𝑠0 ,𝑢0 (𝑠0 + 𝜏) − 𝑔𝑠0 ,𝑢0 (𝑠0 ) − 𝐷1 ℎ(𝑠0 + 𝜃, 𝑢0 )𝜏 𝑠0 +𝜏

− ∫ [𝐷1 ℎ(𝜎, 𝑢0 + 𝜏) − 𝐷1 ℎ(𝜎, 𝑢0 )] 𝑑𝜎 𝑠0

with 0 ≤ 𝜃 ≤ 𝜏. However, the partial derivative 𝐷1 ℎ is continuous by what we have just shown, and so 𝐷2 ℎ(𝑠0 , 𝑢0 ) = lim

𝜏→0

ℎ(𝑠0 , 𝑢0 + 𝜏) − ℎ(𝑠0 , 𝑢0 ) = 𝑔𝑠󸀠0 ,𝑢0 (𝑠0 ) − 𝐷1 ℎ(𝑠0 , 𝑢0 ) . 𝜏

Given 𝜀 > 0, we may choose 𝛿 > 0 such that |𝑠 − 𝑡| ≤ 𝛿 and |𝑢 − 𝑣| ≤ 𝛿 implies 󸀠 󸀠 (𝑠) − 𝑔𝑡,𝑣 (𝑡)| ≤ 𝜀, |𝑔𝑠,𝑢

|𝐷1 ℎ(𝑠, 𝑢) − 𝐷1 ℎ(𝑡, 𝑣)| ≤ 𝜀,

‖𝑔𝑠,𝑢 − 𝑔𝑡,𝑣 ‖𝐶1 ≤ 𝜀

because 𝐷1 ℎ is continuous on [𝑎, 𝑏]× ℝ, the operator 𝑆ℎ is continuous in 𝐶1 ([𝑎, 𝑏]), and ‖𝑓𝑠,𝑢 − 𝑓𝑡,𝑣 ‖𝐶1 = |𝑢 − 𝑣 − 𝑠 + 𝑡| ≤ |𝑢 − 𝑣| + |𝑠 − 𝑡| ≤ 2𝛿 . Combining these estimates, we obtain 󸀠 󸀠 |𝐷2 ℎ(𝑠, 𝑢) − 𝐷2 ℎ(𝑡, 𝑣)| = |𝑔𝑠,𝑢 (𝑠) − 𝐷1 ℎ(𝑠, 𝑢) − 𝑔𝑡,𝑣 (𝑡) + 𝐷1 ℎ(𝑡, 𝑣)| 󸀠 󸀠 󸀠 󸀠 ≤ |𝑔𝑠,𝑢 (𝑠) − 𝑔𝑠,𝑢 (𝑡)| + |𝑔𝑠,𝑢 (𝑡) − 𝑔𝑡,𝑣 (𝑡)| + |𝐷1 ℎ(𝑠, 𝑢) − 𝐷1 ℎ(𝑡, 𝑣)|

≤ 𝜀 + ‖𝑔𝑠,𝑢 − 𝑔𝑡,𝑣 ‖𝐶1 + 𝜀 ≤ 3𝜀 . Therefore, we have shown that the partial derivative 𝐷2 ℎ exists and is continuous on [𝑎, 𝑏] × ℝ as well, and the proof is complete. We may summarize the contents of Theorems 6.1, 6.2, 6.4, and 6.7 in the following form: in addition to the mapping condition 𝑆ℎ (𝑋) ⊆ 𝑋, in case 𝑋 = 𝐶1 , we have to require the continuity of 𝑆ℎ , while in case 𝑋 = 𝐿𝑖𝑝𝛼 , we have to require the bounded­ ness of 𝑆ℎ to obtain a “natural” condition for ℎ. On the other hand, in case 𝑋 = 𝐶 or 𝑋 = 𝐿 𝑝 , we get both the continuity and boundedness of the operator 𝑆ℎ “for free” as a consequence of the mere mapping condition 𝑆ℎ (𝑋) ⊆ 𝑋, which means that in this case, the situation is the same as for the autonomous composition operator (5.1). Our discussion shows that, in contrast to the case of the autonomous composition operator (5.1), almost nothing is known for the nonautonomous superposition opera­ tor (6.1). Therefore, we cannot repeat the complete list of Tables 5.2–5.8 here; the only exception concerns the spaces 𝐿 𝑝 , 𝐶, 𝐿𝑖𝑝𝛼 , and 𝐶1 .

390 | 6 Nonlinear superposition operators Table 6.1. The operator 𝑆ℎ in 𝐿 𝑝 ([𝑎, 𝑏]). 𝑆ℎ bounded in 𝐿 𝑝

⇔

𝑆ℎ (𝐿 𝑝 ) ⊆ 𝐿 𝑝 ⇕ |ℎ(𝑥, 𝑢)| ≤ 𝛼(𝑥) + 𝛽|𝑢|

⇔

𝑆ℎ continuous in 𝐿 𝑝

Table 6.2. The operator 𝑆ℎ in 𝐶([𝑎, 𝑏]). 𝑆ℎ bounded in 𝐶

⇔

𝑆ℎ (𝐶) ⊆ 𝐶 ⇕ ℎ ∈ 𝐶([𝑎, 𝑏] × ℝ)

⇔

𝑆ℎ continuous in 𝐶

Table 6.3. The operator 𝑆ℎ in 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) (0 < 𝛼 ≤ 1). 𝑆ℎ bounded in 𝐿𝑖𝑝𝛼 ⇕ ℎ(⋅, 𝑢) ∈ 𝐿𝑖𝑝𝛼 ℎ(𝑥, ⋅) ∈ 𝐿𝑖𝑝loc (ℝ)

⇒

𝑆ℎ (𝐿𝑖𝑝𝛼 ) ⊆ 𝐿𝑖𝑝𝛼 ⇕ see Proposition 6.5

⇐

𝑆ℎ continuous in 𝐿𝑖𝑝𝛼

Table 6.4. The operator 𝑆ℎ in 𝐶1 ([𝑎, 𝑏]). 𝑆ℎ bounded in 𝐶1

⇒

𝑆ℎ (𝐶1 ) ⊆ 𝐶1

⇐

𝑆ℎ continuous in 𝐶1 ⇕ ℎ ∈ 𝐶1 ([𝑎, 𝑏] × ℝ)

We point out again the differences in Table 6.3 and Table 6.4: in the space 𝐿𝑖𝑝𝛼 , we have a precise characterization of bounded superposition operators, but not for conti­ nuity. In the space 𝐶1 , we have a precise characterization of continuous superposition operators, but not for boundedness. Now, we study the operator (6.1) in the space 𝐵𝑉 which is our main focus. The mixed condition (6.5) for the inclusion 𝑆ℎ (𝐿𝑖𝑝𝛼 ) ⊆ 𝐿𝑖𝑝𝛼 may be rephrased in the fol­ lowing form: – The function ℎ(⋅, 𝑢) : [𝑎, 𝑏] → ℝ is Hölder continuous on [𝑎, 𝑏], uniformly with respect to 𝑢 belonging to compact subsets of ℝ, i.e. sup 𝑙𝑖𝑝𝛼 (ℎ(⋅, 𝑢); [𝑎, 𝑏]) ≤ 𝑣(𝑟) < ∞

|𝑢|≤𝑟

–

for each 𝑟 > 0. The function ℎ(𝑥, ⋅) : ℝ → ℝ is locally Lipschitz continuous on ℝ, uniformly with respect to 𝑥 ∈ [𝑎, 𝑏], i.e. sup |ℎ(𝑥, 𝑢) − ℎ(𝑥, 𝑣)| ≤ 𝑘(𝑟)|𝑢 − 𝑣| (|𝑢|, |𝑣| ≤ 𝑟)

𝑎≤𝑥≤𝑏

for each 𝑟 > 0.

6.1 Boundedness and continuity

| 391

This suggests that the following two conditions are also sufficient for the inclusion 𝑆ℎ (𝐵𝑉) ⊆ 𝐵𝑉. – The function ℎ(⋅, 𝑢) : [𝑎, 𝑏] → ℝ has bounded variation on [𝑎, 𝑏], uniformly with respect to 𝑢 belonging to compact subsets of ℝ, i.e. sup Var(ℎ(⋅, 𝑢); [𝑎, 𝑏]) ≤ 𝑣(𝑟) < ∞

(6.9)

|𝑢|≤𝑟

–

for each 𝑟 > 0. The function ℎ(𝑥, ⋅) : ℝ → ℝ is locally Lipschitz continuous on ℝ, uniformly with respect to 𝑥 ∈ [𝑎, 𝑏], i.e. sup |ℎ(𝑥, 𝑢) − ℎ(𝑥, 𝑣)| ≤ 𝑘(𝑟)|𝑢 − 𝑣|

𝑎≤𝑥≤𝑏

(|𝑢|, |𝑣| ≤ 𝑟)

(6.10)

for each 𝑟 > 0. As a matter of fact, these condition have been formulated, without proof, by Lyamin in [189], and afterwards cited and used by many authors (e.g. in the book [20]). However, it came out as a surprise only quite recently that this is false. Indeed, Maćkowiak [190] has shown this by means of a sophisticated counterexample which looks as follows. Example 6.8. We construct a function ℎ : [0, 1] × ℝ → ℝ which fulfills the two con­ ditions (6.9) and (6.10) as well as a function 𝑓 ∈ 𝐵𝑉([0, 1]) such that 𝑆ℎ 𝑓 ∈ ̸ 𝐵𝑉([0, 1]). For 𝑛 = 2, 3, . . ., let 𝑐𝑛 := 1 −

1 , 𝑛

𝑑𝑛 :=

1 2𝑛

𝐼𝑛 := (𝑐𝑛 − 𝑑𝑛 , 𝑐𝑛 + 𝑑𝑛 ) = (

2𝑛 − 3 2𝑛 − 1 , ). 2𝑛 2𝑛

Clearly, 𝐼𝑛 is a subinterval of (0, 1) of length 1/𝑛. Now, we define ℎ : [0, 1] × ℝ → ℝ by 𝑛| { 𝑛1 (1 − |𝑢−𝑐 ) if 𝑥 = 𝑐𝑛 and 𝑢 ∈ 𝐼𝑛 , 𝑑𝑛 ℎ(𝑥, 𝑢) := { (6.11) 0 otherwise . { Geometrically, the sets {𝑐𝑛 } × 𝐼𝑛 on which ℎ does not vanish are vertical segments without endpoints in the plane which get smaller as 1/𝑛 and are shifted to the right and above as 𝑛 increases. Since

lim

𝑢→𝑐𝑛 ±𝑑𝑛

ℎ(𝑐𝑛 , 𝑢) =

𝑑 1 (1 − 𝑛 ) = 0 , 𝑛 𝑑𝑛

ℎ(𝑥, ⋅) is continuous on ℝ. Even more, the equality sup {|𝐷2 ℎ(𝑐𝑛 , 𝑢)| : 𝑢 ∈ 𝐼𝑛 \ {𝑐𝑛 }} =

1 1 =2 𝑛 𝑑𝑛

shows that the function ℎ(𝑥, ⋅) is (globally) Lipschitz continuous on ℝ, uniformly with respect to 𝑥 ∈ [0, 1], with Lipschitz constant 2, and so (6.10) holds true.

392 | 6 Nonlinear superposition operators Now, we prove that ℎ(⋅, 𝑢) ∈ 𝐵𝑉([0, 1]), uniformly with respect to¹ 𝑢 ∈ [0, 1], and so (6.9) holds true as well. To this end, we fix 𝑚 ∈ ℕ and show that 𝐼𝑚 ∩ 𝐼𝑛 = 0

(𝑛 ≥ 4𝑚) .

(6.12)

In fact, for 𝑛 = 4𝑚, this follows from the inequalities 𝑐𝑛 − 𝑑𝑛 = 1 −

3 1 3 =1− > 1− = 𝑐𝑚 + 𝑑𝑚 . 2𝑛 8𝑚 2𝑚

The function 𝑔 : [1, ∞) → ℝ defined by 𝑔(𝑠) := 1 − 1𝑠 − 2𝑠1 is continuous and strictly increasing, so for 𝑛 > 4𝑚, we get 𝑐𝑛 − 𝑑𝑛 = 𝑔(𝑛) > 𝑔(4𝑚) = 1 −

1 3 1 1 − =1− > 1− = 𝑐𝑚 + 𝑑𝑚 , 4𝑚 8𝑚 8𝑚 2𝑚

which proves (6.12). Since 𝑚 ≥ 2, we further obtain 4𝑚

∑

𝑛=𝑚

1 1 1 1 1 1 1 3𝑚 + 1 = + + ... + ≤ + + ... + < 4. = 𝑛 𝑚 𝑚+1 4𝑚 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑚 𝑚 𝑚 𝑚

(6.13)

3𝑚+1 𝑡𝑒𝑟𝑚𝑠

Now, let 𝑢 ∈ 𝐼𝑚 for some 𝑚 ≥ 2. Then 𝑢 ∈ ̸ 𝐼𝑛 for 𝑛 ≥ 4𝑚, by (6.12). Moreover, (6.13) implies 4𝑚

4𝑞(𝑚)−1

𝑚−1 1 1 1 4𝑚 1 = 2( ∑ + ∑ + ∑ ) ≤ 22 , 𝑛 𝑛 𝑛=4𝑞(𝑚) 𝑛 𝑛=𝑚 𝑛 𝑛=𝑞(𝑚) 𝑛=𝑞(𝑚)

Var(ℎ(⋅, 𝑢); [0, 1]) ≤ 2 ∑

where 𝑞(𝑚) := max {ent(𝑚/4), 2}. Since the upper bound does not depend on 𝑢, the assertion is proved. It remains to show that the operator (6.1) generated by ℎ does not map the space 𝐵𝑉([0, 1]) into itself. Consider the function 𝑓(𝑥) := 𝑥 which trivially belongs to 𝐵𝑉([0, 1]). By definition of ℎ, we then have {1 𝑆ℎ 𝑓(𝑥) = ℎ(𝑥, 𝑥) = { 𝑛 0 {

if 𝑥 = 𝑐𝑛 , otherwise .

Consider a partition of the form 𝑃𝑛 := {0, 𝑡1 , 𝑐2 , 𝑡2 , 𝑐3 , . . . , 𝑐𝑛 , 𝑡𝑛 , 1} ∈ P([0, 1]) with 𝑐𝑗 < 𝑡𝑗 < 𝑐𝑗+1 for 𝑗 = 1, 2, . . . , 𝑛. Then ℎ(𝑡𝑗 , 𝑡𝑗 ) = 0 implies 𝑛

1 → ∞ (𝑛 → ∞) , 𝑘 𝑘=2

Var(𝑆ℎ 𝑓, 𝑃𝑛 ; [0, 1]) ≥ ∑ and so 𝑆ℎ 𝑓 ∈ ̸ 𝐵𝑉([0, 1]).

1 Obviously, this suffices since ℎ(𝑥, 𝑢) ≡ 0 for 𝑢 ∉ [0, 1].

♥

6.1 Boundedness and continuity

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393

Observe that Example 6.8 shows even more: for ℎ given by (6.11), the corresponding superposition operator 𝑆ℎ does not map any function space 𝑋 ⊆ 𝐵𝑉 which contains the identity 𝑓(𝑥) = 𝑥 into itself.² This is of course in sharp contrast to Theorem 5.9 which asserts that in the autonomous case ℎ : ℝ → ℝ, local Lipschitz continuity of ℎ on the real axis is both necessary and sufficient for the operator (5.1) to map any space 𝑋 with 𝐿𝑖𝑝 ⊆ 𝑋 ⊆ 𝐵𝑉 into itself. Now, we are interested in the problem to what extent the above conditions (6.9) and (6.10) are necessary for the inclusion 𝑆ℎ (𝐵𝑉) ⊆ 𝐵𝑉. Of course, since constant func­ tions belong to 𝐵𝑉([𝑎, 𝑏]), from this inclusion, it follows that ℎ(⋅, 𝑢) ∈ 𝐵𝑉([𝑎, 𝑏]) for every 𝑢 ∈ ℝ. The following simple example shows, however, that one cannot expect local Lipschitz continuity of ℎ(𝑥, ⋅) for every 𝑥 ∈ [𝑎, 𝑏]: Example 6.9. Let ℎ0 (𝑢) := min {√|𝑢|, 1} be the seagull function from Example 5.8, and define ℎ : [0, 1] × ℝ → ℝ by {ℎ0 (𝑢) ℎ(𝑥, 𝑢) := { 0 {

if 𝑥 = 0 , if 0 < 𝑥 ≤ 1 .

(6.14)

Then the corresponding operator (6.1) maps the space 𝐵𝑉([0, 1]) into itself since Var(𝑆ℎ 𝑓, 𝑃; [0, 1]) = |ℎ(0, 𝑓(0))| = |𝑓(0)| for 𝑓 ∈ 𝐵𝑉([0, 1]) and every 𝑃 ∈ P([0, 1]). The same calculation shows that ‖𝑓‖𝐵𝑉 ≤ 𝑟 implies ‖𝑆ℎ 𝑓‖𝐵𝑉 ≤ 2√𝑟, and so the operator 𝑆ℎ is bounded in the norm (1.16) of 𝐵𝑉([0, 1]). On the other hand, the function ℎ(0, ⋅) = ℎ0 is certainly not locally Lips­ chitz at zero, and so (6.10) is not true. ♥ Observe that the Lipschitz continuity of the function ℎ(𝑥, ⋅) fails in Example 6.9 only at one point, namely, 𝑥 = 0. Clearly, one could generalize Example 6.9 by fixing points 𝑡1 , 𝑡2 , . . . , 𝑡𝑛 ∈ [0, 1] and considering the function {ℎ𝑖 (𝑢) ℎ(𝑥, 𝑢) := { 0 {

if 𝑥 = 𝑡𝑖 (𝑖 = 1, 2, . . . , 𝑛) , otherwise ,

(6.15)

where ℎ1 , ℎ2 , . . . , ℎ𝑛 are any functions (even discontinuous). In spite of the finitely many “singularities” 𝑡1 , 𝑡2 , . . . , 𝑡𝑛 ∈ [0, 1], the superposition operator 𝑆ℎ defined by (6.15) then maps 𝐵𝑉([0, 1]) into itself and is bounded. The following theorem shows that excluding countably many points 𝑥 ∈ [𝑎, 𝑏] of this type for ℎ(𝑥, ⋅), one gets a necessary condition for 𝑆ℎ to be bounded in 𝐵𝑉([𝑎, 𝑏]). This means, roughly speaking, that the function (6.14) and the more general function (6.15) are in a certain sense “optimal.”

2 This makes Example 6.6 obsolete, but not Example 6.3 because 𝐿𝑖𝑝𝛼 ⊈ 𝐵𝑉 for 𝛼 < 1, see Exam­ ple 1.24.

394 | 6 Nonlinear superposition operators Theorem 6.10. Suppose that the operator (6.1) generated by ℎ : [𝑎, 𝑏] × ℝ → ℝ is bounded in the space 𝐵𝑉([𝑎, 𝑏]). Then the function ℎ may be represented as a sum ̃ 𝑢) + ℎ(𝑥, ̂ 𝑢) , ℎ(𝑥, 𝑢) = ℎ(𝑥,

(6.16)

where ℎ̃ and ℎ̂ have the following properties. (a) The function ℎ̃ satisfies (6.9) and (6.10). ̂ 𝑢) is zero on ([𝑎, 𝑏] \ 𝐶) × ℝ, where 𝐶 ⊂ [𝑎, 𝑏] is some countable (b) The function ℎ(𝑥, subset. (c) The superposition operators 𝑆ℎ̃ and 𝑆ℎ̂ generated by ℎ̃ and ℎ,̂ respectively, are both bounded in 𝐵𝑉([𝑎, 𝑏]). Proof. For the proof, we take [𝑎, 𝑏] = [0, 1] and consider the space 𝐵𝑉([0, 1]) equipped with the norm ⦀𝑓⦀𝐵𝑉 := max {‖𝑓‖∞ , Var(𝑓; [0, 1])} , (6.17) with ‖⋅‖∞ given by (0.39). The norm (6.17) is equivalent to the usual norm (1.16) because ‖𝑓‖𝐵𝑉 ≤ ⦀𝑓⦀𝐵𝑉 ≤ 2‖𝑓‖𝐵𝑉 by (1.9). Suppose that, for any countable set 𝐶 ⊂ [0, 1], the function ℎ(𝑡, ⋅) does not belong to 𝐿𝑖𝑝loc (ℝ) for 𝑡 ∈ [0, 1] \ 𝐶. In particular, we then find 𝑘 ∈ ℕ such that ℎ(𝑡, ⋅) ∈ ̸ 𝐿𝑖𝑝([−𝑘, 𝑘]) for 𝑡 ∈ [0, 1] \ 𝐶. Keeping this 𝑘 fixed for the moment, we conclude that the set 𝐸𝑚,𝑘 := {𝑡 ∈ [0, 1] : |ℎ(𝑡, 𝑢) − ℎ(𝑡, 𝑣)| > 𝑚|𝑢 − 𝑣| for some 𝑢, 𝑣 ∈ [−𝑘, 𝑘]}

(6.18)

is uncountable for all 𝑚 ∈ ℕ. For any 𝑢, 𝑣 ∈ [−𝑘, 𝑘], we may choose points 𝑢𝑗 , 𝑣𝑗 ∈ [−𝑘, 𝑘] (𝑗 = 1, 2, . . . , 10𝑘) such that |𝑢𝑗 − 𝑣𝑗 | ≤ 1/5 and |ℎ(𝑡, 𝑢) − ℎ(𝑡, 𝑣)| ≤ 10𝑘 max {|ℎ(𝑡, 𝑢𝑗 ) − ℎ(𝑡, 𝑣𝑗 )| : 𝑗 = 1, 2, . . . , 10𝑘} for all 𝑡 ∈ [0, 1]. Therefore, for each 𝑚 ∈ ℕ and all 𝑡 ∈ 𝐸𝑚,𝑘 , there are 𝑢, 𝑣 ∈ [−𝑘, 𝑘] such that 𝑚 1 |𝑢 − 𝑣| . |𝑢 − 𝑣| ≤ , |ℎ(𝑡, 𝑢) − ℎ(𝑡, 𝑣)| > 5 10𝑘 Now, we use our assumption that the operator 𝑆ℎ maps 𝐵𝑉([0, 1]) into itself and is bounded. In particular, 𝑆ℎ maps the ball 𝐵̃𝑘 := {𝑓 ∈ 𝐵𝑉([0, 1]) : ⦀𝑓⦀𝐵𝑉 ≤ 𝑘} into some ball 𝐵𝑅 := {𝑓 ∈ 𝐵𝑉([0, 1]) : ‖𝑓‖𝐵𝑉 ≤ 𝑅}. Fix 𝑚 ≥ 10𝑘𝑅𝜌, where 𝜌 = 16, and let E := 𝐸𝑚,𝑘 for this 𝑚, with 𝐸𝑚,𝑘 given by (6.18). For 𝑟 = 5, 6, . . ., denote by E𝑟 the set of all 𝑡 ∈ E such that 𝑢 and 𝑣 can be chosen with 1 1 < |𝑢 − 𝑣| ≤ . 2𝑟 𝑟 Since E is the union of all sets E𝑟 , 𝑟 = 5, 6, . . ., some E𝑟 must be uncountable, and we can assume E = E𝑟 . Choose 𝛿 ∈ (0, 1/9) such that 2𝑟𝛿 = 1, and fix 𝑢 = 𝑥𝑡 and 𝑣 = 𝑦𝑡 such that 𝛿 < 𝑦𝑡 − 𝑥𝑡 < 2𝛿 for all 𝑡. There is a finite set 𝐹 ⊂ [−𝑘, 𝑘] such that each point in [−𝑘, 𝑘] has distance < 𝛿/3 from some point in 𝐹. This means that for some 𝑢 ∈ 𝐹,

6.1 Boundedness and continuity

|

395

there is an uncountable set 𝐴 𝑢 ⊆ [0, 1] with 𝑥𝑡 ≤ 𝑢 ≤ 𝑦𝑡 for all 𝑡 ∈ 𝐴 𝑢 . Consider a partition 𝑃 = {𝑠0 , 𝑠1 , . . . , 𝑠𝑛 , 𝑠𝑛+1 } ∈ P([0, 1]), where 𝑠0 = 0, 𝑠𝑛+1 = 1, 𝑠1 , . . . , 𝑠𝑛 ∈ 𝐴 𝑢 , and 𝑛 is so large that 𝑛𝜌𝛿 > 2. Now, we define a function 𝑔 : [0, 1] → ℝ recursively as follows. First, let 𝑔(𝑠0 ) = 𝑔(0) := 𝑢. Suppose we have defined 𝑔(𝑠0 ), . . . , 𝑔(𝑠𝑗−1 ) for 𝑗 ≤ 𝑛, where 𝑔(𝑠𝑖 ) = 𝑥𝑠𝑖 or 𝑔(𝑠𝑖 ) = 𝑦𝑠𝑖 , for each 𝑖 ∈ {1, 2, . . . , 𝑗−1}, so that |𝑔(𝑠𝑖 )−𝑢| ≤ 2𝛿. Since |ℎ(𝑠𝑗 , 𝑥𝑠𝑗 )−ℎ(𝑠𝑗 , 𝑦𝑠𝑗 )| > 𝑅𝜌𝛿, we can now choose 𝑔(𝑠𝑗 ) = 𝑥𝑠𝑗 or 𝑔(𝑠𝑗 ) = 𝑦𝑠𝑗 in such a way that |ℎ(𝑠𝑗 , 𝑔(𝑠𝑗 )) − ℎ(𝑠𝑗−1 , 𝑔(𝑠𝑗−1 ))| >

1 𝑅𝜌𝛿 . 2

In either case, |𝑔(𝑠𝑗 ) − 𝑔(𝑠𝑗−1 )| ≤ 4𝛿. Thus, we have defined 𝑔(𝑠𝑗 ) for 𝑗 = 0, 1, . . . , 𝑛. Now, for 𝜈 = 1, 2, . . . , 𝑛, we have 𝜈

∑ |ℎ(𝑠𝑗 , 𝑔(𝑠𝑗 )) − ℎ(𝑠𝑗−1 , 𝑔(𝑠𝑗−1 ))| ≥

𝑗=1

1 𝜈𝑅𝜌𝛿 . 2

There is a smallest 𝜈 ∈ {2, 3, . . . , 𝑛} such that 𝜈𝜌𝛿 > 2, hence 8𝜈𝛿 > 1, and for this 𝜈, we have 𝜈

𝜈

∑ |𝑆ℎ 𝑔(𝑠𝑗 ) − 𝑆ℎ 𝑔(𝑠𝑗−1 )| = ∑ |ℎ(𝑠𝑗 , 𝑔(𝑠𝑗 )) − ℎ(𝑠𝑗−1 , 𝑔(𝑠𝑗−1 ))| > 𝑅 .

𝑗=1

(6.19)

𝑗=1

Since 𝑘 ≥ 1 and 𝜈𝜌𝛿 ≤ 4, we further obtain 𝜈

∑ |𝑔(𝑠𝑗 ) − 𝑔(𝑠𝑗−1 )| ≤ 4𝜈𝛿 ≤ 1 ≤ 𝑘 .

(6.20)

𝑗=1

Now, we are almost done. Using the function 𝑔, we define 𝑓 : [0, 1] → ℝ by 𝑔(𝑠𝑗 ) { { { 𝑓(𝑡) := {𝑔(𝑠𝜈 ) { { {linear

for 𝑡 = 𝑠𝑗 , 𝑗 = 0, 2, . . . , 𝜈 , for 𝑠𝜈 ≤ 𝑡 ≤ 1 , otherwise .

By monotonicity, the variation Var(𝑓, 𝑃; [0, 1]) with respect to any partition 𝑃 ∈ P([0, 1]) is dominated by one in which adjoining increments of the same sign are com­ bined, so we can assume that adjoining increments are of opposite signs. Moreover, we may suppose that the partition points are local maxima or minima of 𝑓, and that only the points 𝑠0 , . . . , 𝑠𝜈 occur in the partition. Each value 𝑔(𝑠𝑗 ) has distance ≤ 2𝛿 from 𝑢, so Var(𝑓; [0, 1]) ≤ 4𝜈𝛿 ≤ 1 , by (6.20), and ‖𝑓‖𝐵𝑉 ≤ 1 ≤ 𝑘. Since also ‖𝑓‖∞ ≤ 𝑘, by construction, our definition of the norm (6.17) shows that 𝑓 ∈ 𝐵̃𝑘 . Thus, our choice of 𝑅 implies that 𝑆ℎ 𝑓 ∈ 𝐵𝑅 , i.e.

396 | 6 Nonlinear superposition operators Var(𝑆ℎ 𝑓; [0, 1]) ≤ ‖𝑆ℎ 𝑓‖𝐵𝑉 ≤ 𝑅, contradicting (6.19). This contradiction shows that our assumption was false, and so the assertions (a) and (b) are proved. It remains to prove (c). Since the operator 𝑆ℎ is bounded in the space 𝐵𝑉([0, 1]), by assumption, the ball {𝑓 ∈ 𝐵𝑉([0, 1]) : ‖𝑓‖∞ ≤ ⦀𝑓⦀𝐵𝑉 ≤ 𝑟} is mapped by the operator 𝑆ℎ̃ into a ball {𝑔 ∈ 𝐵([0, 1]) : ‖𝑔‖∞ ≤ 𝑅}, and so 𝑆ℎ̃ is bounded in the space 𝐵([0, 1]). Any variation sum 𝑚

̃ , 𝑓(𝑠 )) − ℎ(𝑠 ̃ Var(𝑆ℎ̃ 𝑓, 𝑃; [0, 1]) = ∑ |ℎ(𝑠 𝑗 𝑗 𝑗−1 , 𝑓(𝑠𝑗−1 ))| 𝑗=1

with respect to some partition 𝑃 = {𝑠0 , 𝑠1 , . . . , 𝑠𝑚 } ∈ P([0, 1]) can be approximated arbitrarily closely by another variation sum 𝑚

̃ , 𝑓(𝑡 )) − ℎ(𝑡 ̃ Var(𝑆ℎ̃ 𝑓, 𝑄; [0, 1]) = ∑ |ℎ(𝑡 𝑗 𝑗 𝑗−1 , 𝑓(𝑡𝑗−1 ))| 𝑗=1

with respect to some partition 𝑄 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]), where 𝑡𝑗 does not belong to the exceptional set 𝐶 from (b), 𝑡𝑗 is close enough to 𝑠𝑗 , and 𝑡𝑗 > 𝑠𝑗 except if 𝑠𝑚 = 1, when 𝑡𝑚 < 𝑠𝑚 . For any such partition 𝑄, there is a function 𝑔 ∈ 𝐵𝑉([0, 1]) with ⦀𝑔⦀𝐵𝑉 ≤ ⦀𝑓⦀𝐵𝑉 and 𝑔(𝑡𝑗 ) = 𝑓(𝑠𝑗 ) for 𝑗 = 1, 2, . . . , 𝑚. It follows that sup {‖𝑆ℎ̃ 𝑓‖𝐵𝑉 : ⦀𝑓⦀𝐵𝑉 ≤ 𝑟} ≤ sup {‖𝑆ℎ 𝑔‖𝐵𝑉 : ⦀𝑔⦀𝐵𝑉 ≤ 𝑟} < ∞ , which shows that the superposition operator 𝑆ℎ̃ is in fact bounded in 𝐵𝑉([0, 1]). The same then holds for the operator 𝑆ℎ̂ = 𝑆ℎ − 𝑆ℎ̃ , and the proof is complete. In view of Example 6.8, the question arises how to strengthen³ (6.9) or (6.10) to get a sufficient condition for the inclusion 𝑆ℎ (𝐵𝑉) ⊆ 𝐵𝑉. An interesting condition was proposed by Bugajewska [66]; to state this condition, we need some notation. Given 𝑟 > 0, by U𝑟 , we denote the family of all finite collections 𝑈𝑟 = {𝑢1 , 𝑢2 , . . . , 𝑢𝑚 } ⊂ [−𝑟, 𝑟]; thus, the family U𝑟 may be regarded as some kind of “vertical” analogue of the partition family P([𝑎, 𝑏]) in the “horizontal” direction. For ℎ : [𝑎, 𝑏] × ℝ → ℝ, 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([𝑎, 𝑏]), and 𝑈𝑟 = {𝑢1 , 𝑢2 , . . . , 𝑢𝑚 } ∈ U𝑟 , we put 𝑚

Var(ℎ, 𝑃, 𝑈𝑟 ; [𝑎, 𝑏] × ℝ) := ∑ |ℎ(𝑡𝑗 , 𝑢𝑗 ) − ℎ(𝑡𝑗−1 , 𝑢𝑗 )|

(6.21)

𝑗=1

and Var(ℎ, 𝑈𝑟 ; [𝑎, 𝑏] × ℝ) := sup {Var(ℎ, 𝑃, 𝑈𝑟 ; [𝑎, 𝑏] × ℝ) : 𝑃 ∈ P([𝑎, 𝑏])}

(6.22)

3 Since (6.10) is necessary and sufficient in the autonomous case, it seems reasonable to only strengthen condition (6.9).

6.1 Boundedness and continuity |

397

where the supremum in (6.22) is taken over all partitions 𝑃 ∈ P([𝑎, 𝑏]). Using this notation, we now replace (6.9) by the stronger condition⁴ sup Var(ℎ, 𝑈𝑟 ; [𝑎, 𝑏] × ℝ) ≤ 𝑣(𝑟) < ∞

(6.23)

𝑈𝑟 ∈U𝑟

for each 𝑟 > 0. The next theorem [66] shows that this condition, together with (6.10), in fact, suffices to guarantee what we need. Theorem 6.11. Suppose that ℎ satisfies the conditions (6.10) and (6.23). Then the corre­ sponding operator (6.1) maps the space 𝐵𝑉([𝑎, 𝑏]) into itself and is bounded. Proof. Without loss of generality, we take [𝑎, 𝑏] = [0, 1]. Fix 𝑓 ∈ 𝐵𝑉([0, 1]) with ‖𝑓‖𝐵𝑉 ≤ 𝑟, 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]), and 𝑈𝑟 = {𝑢1 , 𝑢2 , . . . , 𝑢𝑚 } ∈ U𝑟 , where 𝑢𝑗 = 𝑓(𝑡𝑗 ) (𝑗 = 1, 2, . . . , 𝑚). Then 𝑚

Var(𝑆ℎ 𝑓, 𝑃; [0, 1]) = ∑ |ℎ(𝑡𝑗 , 𝑓(𝑡𝑗 )) − ℎ(𝑡𝑗−1 , 𝑓(𝑡𝑗−1 ))| 𝑗=1 𝑚

𝑚

≤ ∑ |ℎ(𝑡𝑗 , 𝑓(𝑡𝑗 )) − ℎ(𝑡𝑗−1 , 𝑓(𝑡𝑗 ))| + ∑ |ℎ(𝑡𝑗−1 , 𝑓(𝑡𝑗 )) − ℎ(𝑡𝑗−1 , 𝑓(𝑡𝑗−1 ))| 𝑗=1

𝑗=1

𝑚

≤ 𝑣(𝑟) + 𝑘(𝑟) ∑ |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| ≤ 𝑣(𝑟) + 𝑘(𝑟)𝑟 , 𝑗=1

where 𝑣(𝑟) is taken from (6.23) and 𝑘(𝑟) from (6.10). This not only shows that 𝑆ℎ 𝑓 be­ longs to 𝐵𝑉([0, 1]), but also the boundedness of 𝑆ℎ in the norm (1.16). Observe that for the function (6.14) in Example 6.9, we have 𝑚

∑ |ℎ(𝑡𝑗 , 𝑢𝑗 ) − ℎ(𝑡𝑗−1 , 𝑢𝑗 )| = |ℎ(0, 𝑢1 )| = min {√|𝑢1 |, 1} ,

𝑗=1

and so (6.23) holds in this example with 𝑣(𝑟) := min {√𝑟, 1}. Here is another example which illustrates Theorem 6.11. Example 6.12. Suppose that ℎ factorizes into two functions of the form ℎ(𝑥, 𝑢) = ℎ1 (𝑥)ℎ2 (𝑢), where ℎ1 ∈ 𝐵𝑉([𝑎, 𝑏]) and ℎ2 ∈ 𝐿𝑖𝑝loc (ℝ). In this case, (6.10) is trivially true since |ℎ(𝑥, 𝑢) − ℎ(𝑥, 𝑣)| = |ℎ1 (𝑥)| |ℎ2 (𝑢) − ℎ2 (𝑣)| ≤ ‖ℎ1 ‖∞ 𝑙𝑖𝑝(ℎ2 ; [−𝑟, 𝑟])|𝑢 − 𝑣| .

4 Indeed, if we restrict ourselves to singletons 𝑈𝑟 = {𝑢} ∈ U𝑟 with |𝑢| ≤ 𝑟 in (6.23), we exactly get condition (6.9).

398 | 6 Nonlinear superposition operators However, (6.23) is also true because for 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]) and 𝑈𝑟 = {𝑢1 , 𝑢2 , . . . , 𝑢𝑚 } ∈ U𝑟 , we have 𝑚

𝑚

∑ |ℎ(𝑡𝑗 , 𝑢𝑗 ) − ℎ(𝑡𝑗−1 , 𝑢𝑗 )| = ∑ |ℎ1 (𝑡𝑗 ) − ℎ1 (𝑡𝑗−1 )| |ℎ2 (𝑢𝑗 )|

𝑗=1

𝑗=1

≤ sup |ℎ2 (𝑢)| Var(ℎ1 ; [𝑎, 𝑏]) . |𝑢|≤𝑟

This case occurs frequently in applications. We point out again that the corre­ sponding condition (6.9) in this case is not sufficient, as Example 6.9 (with ℎ1 := 𝜒{0} and ℎ2 := ℎ0 ) shows. ♥ We may summarize our discussion on necessary or sufficient conditions for the inclu­ sion 𝑆ℎ (𝐵𝑉) ⊆ 𝐵𝑉 as follows: – Conditions (6.9) and (6.10) are not sufficient for 𝑆ℎ (𝐵𝑉) ⊆ 𝐵𝑉, but conditions (6.10) and (6.23) are. – Condition (6.9) is necessary for 𝑆ℎ (𝐵𝑉) ⊆ 𝐵𝑉, but condition (6.10) is not. Now, we state a generalization of Theorem 6.11 to the space 𝑊𝐵𝑉𝜙 , see Section 2.1, which is also due to Bugajewska [66]. To this end, we suppose that 𝜙 : [0, ∞) → [0, ∞) is a Young function which satisfies the 𝛿2 -condition introduced in Definition 2.4. As a consequence, the function 𝑀 : (0, ∞) × [1, ∞) → [1, ∞) defined by 𝑀(𝑇, 𝜆) = sup {

𝜙(𝜆𝑡) : 0 < 𝑡 ≤ 𝑇} 𝜙(𝑡)

(6.24)

is well-defined and increasing⁵ in both 𝑇 and 𝜆. Observe that for 𝑓 ∈ 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) and 𝜆 ≥ 1, we then have 𝑊 Var𝑊 𝜙 (𝜆𝑓; [𝑎, 𝑏]) ≤ 𝑀(2‖𝑓‖∞ , 𝜆) Var𝜙 (𝑓; [𝑎, 𝑏]) ,

(6.25)

by the Definition (2.2) of the Wiener–Young variation. We also have to replace (6.21) and (6.22) by 𝑚

Var𝑊 𝜙 (ℎ, 𝑃, 𝑈𝑟 ; [𝑎, 𝑏] × ℝ) := ∑ 𝜙 (|ℎ(𝑡𝑗 , 𝑢𝑗 ) − ℎ(𝑡𝑗−1 , 𝑢𝑗 )|)

(6.26)

𝑗=1

and 𝑊 Var𝑊 𝜙 (ℎ, 𝑈𝑟 ; [𝑎, 𝑏] × ℝ) := sup {Var𝜙 (ℎ, 𝑃, 𝑈𝑟 ; [𝑎, 𝑏] × ℝ) : 𝑃 ∈ P([𝑎, 𝑏])}

(6.27)

respectively, where the supremum in (6.27) is again taken over all partitions 𝑃 ∈ P([𝑎, 𝑏]).

5 Note that 𝑀(𝑇, 2) coincides with the characteristic 𝑀(𝑇) which we introduced in (2.6). In the Wiener space 𝑊𝐵𝑉𝑝 , we have the explicit formula 𝑀(𝑇, 𝜆) = 𝜆𝑝 for every 𝑇 > 0.

6.1 Boundedness and continuity

| 399

Theorem 6.13. Suppose that ℎ satisfies (6.10) and sup Var𝑊 𝜙 (ℎ, 𝑈𝑟 ; [𝑎, 𝑏] × ℝ) ≤ 𝑣𝜙 (𝑟) < ∞

(6.28)

𝑈𝑟 ∈U𝑟

for each 𝑟 > 0, where 𝜙 ∈ 𝛿2 is a Young function, and Var𝑊 𝜙 (ℎ, 𝑈𝑟 ; [𝑎, 𝑏] × ℝ) is defined by (6.27). Then the corresponding operator (6.1) maps the space 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) into itself and is bounded in the norm (2.11). Proof. We take again [𝑎, 𝑏] = [0, 1]. Fix 𝑓 ∈ 𝑊𝐵𝑉𝜙 ([0, 1]) with ‖𝑓‖𝑊𝐵𝑉𝜙 ≤ 𝑟, 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]), and 𝑈𝑟 = {𝑢1 , 𝑢2 , . . . , 𝑢𝑚 } ∈ U𝑟 . For ̂ := max {𝑘(𝑟), 1}, 𝑘(𝑟)

̂ 𝑇 := max {𝑣𝜙 (𝑟), 2𝑟𝑘(𝑟)},

we then obtain 𝜙(|ℎ(𝑡𝑗 , 𝑢𝑗 ) − ℎ(𝑡𝑗−1 , 𝑢𝑗−1 )|) ≤ 𝜙(|ℎ(𝑡𝑗 , 𝑢𝑗 ) − ℎ(𝑡𝑗−1 , 𝑢𝑗 )| + |ℎ(𝑡𝑗−1 , 𝑢𝑗 ) − ℎ(𝑡𝑗−1 , 𝑢𝑗−1 )|) 1 ≤ 𝜙 (2|ℎ(𝑡𝑗 , 𝑢𝑗 ) − ℎ(𝑡𝑗−1 , 𝑢𝑗 )| + 2𝑘(𝑟)|𝑢𝑗 − 𝑢𝑗−1 |) 2 ̂ 𝑀(𝑇, 4𝑘(𝑟)) ≤ [𝜙 (|ℎ(𝑡𝑗 , 𝑢𝑗 ) − ℎ(𝑡𝑗−1 , 𝑢𝑗 )|) + 𝜙 (|𝑢𝑗 − 𝑢𝑗−1 |)] , 2 (6.29) where we have used the monotonicity and convexity of 𝜙 as well as (6.10) and (6.25). Now, putting 𝑢𝑗 = 𝑓(𝑡𝑗 ) and 𝑢𝑗−1 = 𝑓(𝑡𝑗−1 ) in (6.29) and summing up over 𝑗 = 1, 2, . . . , 𝑚, we further get Var𝑊 𝜙 (𝑆ℎ 𝑓, 𝑃; [0, 1]) 𝑚

= ∑ 𝜙(|ℎ(𝑡𝑗 , 𝑓(𝑡𝑗 )) − ℎ(𝑡𝑗−1 , 𝑓(𝑡𝑗−1 ))|) 𝑗=1

≤

𝑚 ̂ 𝑀(𝑇, 4𝑘(𝑟)) ∑ [𝜙 (|ℎ(𝑡𝑗 , 𝑓(𝑡𝑗 )) − ℎ(𝑡𝑗−1 , 𝑓(𝑡𝑗 ))|) + 𝜙 (|𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )|)] 2 𝑗=1

̂ ̂ 𝑀(𝑇, 4𝑘(𝑟)) 𝑀(𝑇, 4𝑘(𝑟)) [𝑣𝜙 (𝑟) + Var𝑊 (𝑣𝜙 (𝑟) + 𝑟) . 𝜙 (𝑓; [0, 1])] ≤ 2 2 This not only shows that 𝑆ℎ 𝑓 ∈ 𝑊𝐵𝑉𝜙 ([0, 1]), but also proves the boundedness of 𝑆ℎ on balls of radius 𝑟 ≤ 1. To prove the boundedness of 𝑆ℎ on larger balls, we use a rescaling argument and the 𝛿2 -property of 𝜙. Let 𝑓 ∈ 𝑊𝐵𝑉𝜙 ([0, 1]) with ‖𝑓‖𝑊𝐵𝑉𝜙 ≤ 𝑟, and hence ‖𝑓‖∞ ≤ 𝑟 for some 𝑟 > 1. Then (2.11) and (6.25) imply ≤

𝑟𝑓 𝑓 ; [0, 1]) ≤ 𝑀(2𝑟, 𝑟) Var𝑊 𝜙 ( ; [0, 1]) 𝑟 𝑟 󵄩󵄩 𝑓 󵄩󵄩 𝑟 󵄩󵄩 󵄩󵄩 ≤ 𝑀(2𝑟, 𝑟) = 𝑀(2𝑟, 𝑟) ≤ 𝑀(2𝑟, 𝑟) 󵄩󵄩 󵄩󵄩 󵄩󵄩 𝑟 󵄩󵄩𝑊𝐵𝑉𝜙 𝑟

𝑊 Var𝑊 𝜙 (𝑓; [0, 1]) = Var𝜙 (

since ‖𝑓/𝑟‖𝑊𝐵𝑉𝜙 ≤ 1. In this way, we have reduced the case 𝑟 > 1 to the previous case, and the proof is complete.

400 | 6 Nonlinear superposition operators We do not know whether or not Theorem 6.13 is also true without the requirement 𝜙 ∈ 𝛿2 .

6.2 Lipschitz continuity Suppose that the superposition operator 𝑆ℎ given by (6.1) maps a normed space 𝑋 into a normed space 𝑌. In this section, we are going to study the (global) Lipschitz condi­ tion ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖𝑌 ≤ 𝐾‖𝑓 − 𝑔‖𝑋 (𝑓, 𝑔 ∈ 𝑋) , (6.30) compare it with the (local) Lipschitz condition ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖𝑌 ≤ 𝐾(𝑟)‖𝑓 − 𝑔‖𝑋

(𝑓, 𝑔 ∈ 𝑋, ‖𝑓‖𝑋 , ‖𝑔‖𝑋 ≤ 𝑟) ,

(6.31)

and try to develop a parallel theory to that given in Sections 5.4 and 5.5 for the au­ tonomous composition operator (5.1). Clearly, if 𝑌 is an algebra, every affine function ℎ(𝑥, 𝑢) = 𝛼(𝑥) + 𝛽(𝑥)𝑢

(𝛼, 𝛽 ∈ 𝑌)

(6.32)

generates a superposition operator 𝑆ℎ which satisfies (6.30) with 𝐾 := ‖𝛽‖𝑌 . Con­ versely, as mentioned before, from (6.30), it follows in many function spaces 𝑌 that the function ℎ must be of the form (6.32). We have encountered this degeneracy phe­ nomenon in Section 5.4 and describe such spaces by the Matkowski property. Fortunately, there are important function spaces which do not have the Matkowski property. Two examples of such spaces are contained in the following Theorems 6.14 and 6.15; the proofs of these theorems may be found in the paper [9]. Theorem 6.14. Suppose that the operator (6.1) maps the space 𝐶([𝑎, 𝑏]) into itself. Then this operator satisfies (6.30) if and only if the corresponding function ℎ satisfies |ℎ(𝑥, 𝑢) − ℎ(𝑥, 𝑣)| ≤ 𝑔(𝑥)|𝑢 − 𝑣| (𝑎 ≤ 𝑥 ≤ 𝑏, 𝑢, 𝑣 ∈ ℝ)

(6.33)

for some continuous nonnegative function 𝑔 : [𝑎, 𝑏] → ℝ. Theorem 6.15. Suppose that the operator (6.1) maps the space 𝐿 𝑝 ([𝑎, 𝑏]) into the space 𝐿 𝑞 ([𝑎, 𝑏]), where 1 ≤ 𝑞 ≤ 𝑝 < ∞. Then this operator satisfies (6.30) if and only if the corresponding function ℎ satisfies |ℎ(𝑥, 𝑢) − ℎ(𝑥, 𝑣)| ≤ 𝑔(𝑥, 𝑤)|𝑢 − 𝑣| (𝑎 ≤ 𝑥 ≤ 𝑏, |𝑢|, |𝑣| ≤ 𝑤) ,

(6.34)

where the superposition operator 𝑆𝑔 defined by the nonnegative function 𝑔 maps 𝐿 𝑝 ([𝑎, 𝑏]) into 𝐿 𝑝𝑞/(𝑝−𝑞) ([𝑎, 𝑏]). In particular, in case 𝑝 = 𝑞, we have 𝑆𝑔 (𝐿 𝑝 ) ⊆ 𝐿 ∞ , and so condition (6.34) is equivalent to (6.33).

6.2 Lipschitz continuity

| 401

We remark that necessary and sufficient conditions under which the operator (6.1) fulfills the hypotheses of these theorems are well known, see Theorems 6.1 and 6.2 in the first section. Clearly, the Lipschitz condition (6.33) may be replaced by the simpler condition |ℎ(𝑥, 𝑢) − ℎ(𝑥, 𝑣)| ≤ 𝑘|𝑢 − 𝑣| (𝑎 ≤ 𝑥 ≤ 𝑏, 𝑢, 𝑣 ∈ ℝ) ,

(6.35)

where 𝑘 := ‖𝑔‖𝐶 in Theorem 6.14, and 𝑘 := ‖𝑔‖𝐿 ∞ in Theorem 6.15. In the next propo­ sition, we give a very mild condition on two spaces 𝑋 and 𝑌 which implies that the function ℎ satisfies (6.35) whenever the corresponding operator 𝑆ℎ satisfies the Lips­ chitz condition (6.30). Proposition 6.16. Suppose that the operator (6.1) maps a normed space 𝑋 into a normed space 𝑌 and satisfies the Lipschitz condition (6.30). Assume that the space 𝑋 contains the constant functions, and the space 𝑌 is imbedded into the space of bounded functions. Then the function ℎ satisfies the Lipschitz condition (6.35). Proof. The proof is very simple. From (6.30) and our hypothesis 𝑌 󳨅→ 𝐵([𝑎, 𝑏]), it fol­ lows that ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖∞ ≤ 𝐾𝑐(𝑌, 𝐵)‖𝑓 − 𝑔‖𝑋 (𝑓, 𝑔 ∈ 𝑋) , (6.36) where ‖⋅‖∞ denotes the norm (0.39) and 𝑐(𝑌, 𝐵) denotes the imbedding constant (0.36). Choosing 𝑓(𝑥) ≡ 𝑢 and 𝑔(𝑥) ≡ 𝑣 in (6.36) yields |ℎ(𝑥, 𝑢) − ℎ(𝑥, 𝑣)| ≤ ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖∞ ≤ 𝐾𝑐(𝑌, 𝐵)‖𝑓1 ‖𝑋 |𝑢 − 𝑣| , where 𝑓1 denotes the constant function 𝑓1 (𝑥) ≡ 1. So, (6.35) holds with 𝑘 := 𝐾𝑐(𝑌, 𝐵) ‖𝑓1 ‖𝑋 . A typical example for applying Proposition 6.16 is 𝑋 = 𝑌 = 𝐶([𝑎, 𝑏]); in this case, we have 𝑐(𝐶, 𝐵) = 1 and ‖𝑓1 ‖𝐶 = 1, and thus 𝑘 = 𝐾. On the other hand, in case 𝑋 = 𝑌 = 𝐶1 ([𝑎, 𝑏]), say, Proposition 6.16 is too “rough;” in fact, one may prove that (6.30) in this case leads to affine functions ℎ, not just Lipschitz continuous functions ℎ. To show this, we need a refinement of Proposition 6.16 which is based on the use of special polynomials rather than constant functions [226]. This refinement covers several spaces with the Matkowski property, as we shall see. Without loss of generality, we take [𝑎, 𝑏] = [0, 1]. By 𝑃𝑛 ([0, 1]), we denote the linear space of all polynomials of degree ≤ 𝑛; in particular, 𝑃1 ([0, 1]) is the space of all affine functions. We consider 𝑃𝑛 ([0, 1]) equipped with the 𝐶𝑛 -norm (0.63); in particular, the polynomial 𝑓(𝑥) = 𝐴𝑥+𝐵 has the norm ‖𝑓‖𝐶1 = |𝑓(0)| + ‖𝑓󸀠 ‖ = |𝐴| + |𝐵|. Fix 𝑥1 , 𝑥2 ∈ [0, 1] and 𝑢1 , 𝑢2 ∈ ℝ, where 𝑥1 ≠ 𝑥2 , and define 𝑓 ∈ 𝑃1 ([0, 1]) by 𝑓(𝑥) :=

𝑢1 − 𝑢2 𝑥 𝑢 − 𝑥2 𝑢1 𝑢1 (𝑥 − 𝑥2 ) + 𝑢2 (𝑥1 − 𝑥) 𝑥+ 1 2 = . 𝑥1 − 𝑥2 𝑥1 − 𝑥2 𝑥1 − 𝑥2

(6.37)

402 | 6 Nonlinear superposition operators It is not hard to see that this polynomial satisfies the conditions 𝑓(𝑥1 ) = 𝑢1 ,

𝑓(𝑥2 ) = 𝑢2 ,

‖𝑓‖𝐶1 =

|𝑢1 − 𝑢2 | + |𝑥1 𝑢2 − 𝑥2 𝑢1 | . |𝑥1 − 𝑥2 |

Denoting by 𝑔 the analogous polynomial with 𝑢1 replaced by 𝑣1 and 𝑢2 replaced by 𝑣2 in (6.37), i.e. 𝑔(𝑥) :=

𝑥 𝑣 − 𝑥2 𝑣1 𝑣1 (𝑥 − 𝑥2 ) + 𝑣2 (𝑥1 − 𝑥) 𝑣1 − 𝑣2 𝑥+ 1 2 = , 𝑥1 − 𝑥2 𝑥1 − 𝑥2 𝑥1 − 𝑥2

we have 𝑔(𝑥1 ) = 𝑣1 ,

𝑔(𝑥2 ) = 𝑣2 ,

‖𝑔‖𝐶1 =

(6.38)

|𝑣1 − 𝑣2 | + |𝑥1 𝑣2 − 𝑥2 𝑣1 | . |𝑥1 − 𝑥2 |

Consequently, ‖𝑓 − 𝑔‖𝐶1 =

|𝑢1 − 𝑢2 − 𝑣1 + 𝑣2 | + |𝑥1 (𝑢2 − 𝑣2 ) − 𝑥2 (𝑢1 − 𝑣1 )| . |𝑥1 − 𝑥2 |

Moreover, fixing 𝑥 ∈ [0, 1], multiplying by |𝑥1 −𝑥2 |, and letting 𝑥1 → 𝑥 and 𝑥2 → 𝑥, we obtain (6.39) lim |𝑥1 − 𝑥2 |‖𝑓 − 𝑔‖𝐶1 = (1 + |𝑥|)|𝑢1 − 𝑢2 − 𝑣1 + 𝑣2 | . 𝑥1 ,𝑥2 →𝑥

Using this construction, we are now in a position to characterize many spaces with the Matkowski property. Theorem 6.17. Let 𝑋 and 𝑌 be two function spaces over [𝑎, 𝑏]. Assume that the space 𝑃1 ([𝑎, 𝑏]) of affine functions with the 𝐶1 -norm (0.65) is imbedded into 𝑋, and 𝑌 is imbed­ ded into the space 𝐿𝑖𝑝([𝑎, 𝑏]) with norm (0.70). Then (𝑋, 𝑌) has the Matkowski property. Proof. Without loss of generality, we take [𝑎, 𝑏] = [0, 1]. Suppose that the superposi­ tion operator (6.1) maps 𝑋 into 𝑌 and satisfies the global Lipschitz condition (6.30). From our hypotheses 𝑃1 ([0, 1]) 󳨅→ 𝑋 and 𝑌 󳨅→ 𝐿𝑖𝑝([0, 1]), it follows that ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖𝐿𝑖𝑝 ≤ 𝐿‖𝑓 − 𝑔‖𝐶1

(𝑓, 𝑔 ∈ 𝑃1 ([0, 1]))

(6.40)

for 𝐿 := 𝐾𝑐(𝑃1 , 𝑋)𝑐(𝑌, 𝐿𝑖𝑝). In particular, since constant functions belong to 𝑃1 ([0, 1]), we see that ℎ(⋅, 𝑢) ∈ 𝐿𝑖𝑝([0, 1]) for each 𝑢 ∈ ℝ, and so the function ℎ(⋅, 𝑢) is continuous. Fix 𝑥1 , 𝑥2 ∈ [0, 1] and 𝑢1 , 𝑢2 , 𝑣1 , 𝑣2 ∈ ℝ, where 𝑥1 ≠ 𝑥2 , and define 𝑓, 𝑔 ∈ 𝑃1 ([0, 1]) as in (6.37) and (6.38), respectively. By definition (0.70) of the norm in 𝐿𝑖𝑝([0, 1]), we get the estimates 󵄨󵄨 ℎ(𝑥 , 𝑢 ) − ℎ(𝑥 , 𝑢 ) − ℎ(𝑥 , 𝑣 ) + ℎ(𝑥 , 𝑣 ) 󵄨󵄨 󵄨󵄨 1 1 2 2 1 1 2 2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑥1 − 𝑥2 󵄨󵄨 ℎ(𝑥 , 𝑓(𝑥 )) − ℎ(𝑥 , 𝑓(𝑥 )) − ℎ(𝑥 , 𝑔(𝑥 )) + ℎ(𝑥 , 𝑔(𝑥 )) 󵄨󵄨 󵄨 1 1 2 2 1 1 2 2 󵄨󵄨 = 󵄨󵄨󵄨 󵄨󵄨 𝑥1 − 𝑥2 󵄨󵄨 󵄨󵄨 ≤ 𝑙𝑖𝑝(𝑆ℎ 𝑓 − 𝑆ℎ 𝑔) ≤ ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖𝐿𝑖𝑝 ≤ 𝐿‖𝑓 − 𝑔‖𝐶1 ,

6.2 Lipschitz continuity

|

403

by (6.40). Consequently, |ℎ(𝑥1 , 𝑢1 ) − ℎ(𝑥2 , 𝑢2 ) − ℎ(𝑥1 , 𝑣1 ) + ℎ(𝑥2 , 𝑣2 )| ≤ 𝐿|𝑥1 − 𝑥2 |‖𝑓 − 𝑔‖𝐶1 . Fixing now 𝑥 ∈ [0, 1] and letting 𝑥1 → 𝑥 and 𝑥2 → 𝑥, we obtain, by (6.39), |ℎ(𝑥, 𝑢1 ) − ℎ(𝑥, 𝑢2 ) − ℎ(𝑥, 𝑣1 ) + ℎ(𝑥, 𝑣2 )| ≤ 𝐿(1 + |𝑥|)|𝑢1 − 𝑢2 − 𝑣1 + 𝑣2 | .

(6.41)

Substituting 𝑢1 := 𝑦 + 𝑧, 𝑢2 := 𝑦, 𝑣1 := 𝑧 and 𝑣2 := 0, and observing that the right-hand side of (6.41) then becomes zero, we arrive at the equality ℎ(𝑥, 𝑦 + 𝑧) − ℎ(𝑥, 𝑦) − ℎ(𝑥, 𝑧) = −𝛼(𝑥) ,

(6.42)

where we have used the shortcut 𝛼(𝑥) := ℎ(𝑥, 0). Assume first that 𝛼(𝑥) ≡ 0. Then (6.42) shows that for each 𝑥 ∈ [0, 1], the function ℎ(𝑥, ⋅) satisfies the Cauchy functional equation. Moreover, putting 𝑣1 = 𝑣2 = 0 in (6.41), we see that ℎ(𝑥, ⋅) is (Lipschitz) continuous. We conclude that ℎ(𝑥, 𝑢) = 𝛽(𝑥)𝑢 for some function 𝛽 : [0, 1] → ℝ. In ̃ 𝑢) := the general case when 𝛼(𝑥) ≢ 0, we pass from ℎ to the function ℎ̃ defined by ℎ(𝑥, ℎ(𝑥, 𝑢) − 𝛼(𝑥), and the statement follows. The assertion 𝛼, 𝛽 ∈ 𝑌 follows from the fact that 𝛼(𝑥) = ℎ(𝑥, 0) and 𝛽(𝑥) = ℎ(𝑥, 1) − ℎ(𝑥, 0). Theorem 6.17 applies to several spaces we have considered so far. In fact, if all func­ tions in a space 𝑋 have bounded derivatives, they are also Lipschitz continuous. Com­ bining this argument with Proposition 0.40 and Exercises 0.46, 2.31, and 3.53 leads to the following Corollary 6.18. The spaces 𝐿𝑖𝑝, 𝐶1 , 𝐿𝑖𝑝1 𝐿𝑖𝑝1𝛼 , 𝐵𝑉1 , 𝑅𝐵𝑉1𝑝 , 𝑊𝐵𝑉1𝑝 and 𝐴𝐶1 have the Matkowski property. Corollary 6.18 contains Matkowski’s degeneracy results for the space 𝐿𝑖𝑝 proved in [195] and for the space 𝐶1 proved in [196]. However, there are some spaces with the Matkowski property which are not covered by Theorem 6.17 since the requirement 𝑌 󳨅→ 𝐿𝑖𝑝 is too restrictive. For instance, the spaces 𝐿𝑖𝑝𝛼 (𝛼 < 1) and 𝐴𝐶 are excluded by this condition, but do have the Matkowski property, as was shown in [193] and [124], respectively. Theorem 6.17 has another interesting consequence. Suppose that the operator 𝑆ℎ maps the space 𝐶𝑛 ([𝑎, 𝑏]) into the space 𝐶𝑚 ([𝑎, 𝑏]), where 𝑚 > 𝑛, and satisfies (6.40) with ‖ ⋅ ‖𝐿𝑖𝑝 replaced by ‖ ⋅ ‖𝐶𝑚 . Then our proof shows that the function ℎ does not depend on the second variable, which means that the operator 𝑆ℎ is constant. This strong degeneracy phenomenon has been proved directly by Matkowski in [196]. Now, we study Lipschitz continuous superposition operators in the space 𝐵𝑉([𝑎, 𝑏]) of functions of bounded variation. Here, the degeneracy one encounters is some­ what different. Recall that given a function 𝑓 : [𝑎, 𝑏] → ℝ, the right regularization 𝑓# : [𝑎, 𝑏] → ℝ of 𝑓 is defined by {lim𝑠→𝑥+ 𝑓(𝑠) 𝑓#(𝑥) := { 𝑓(𝑏) {

for 𝑎 ≤ 𝑥 < 𝑏 , for 𝑥 = 𝑏 ,

(6.43)

404 | 6 Nonlinear superposition operators while the left regularization 𝑓♭ : [𝑎, 𝑏] → ℝ of 𝑓 is defined by {lim𝑠→𝑥− 𝑓(𝑠) 𝑓♭ (𝑥) := { 𝑓(𝑎) {

for 𝑎 < 𝑥 ≤ 𝑏 , for 𝑥 = 𝑎 .

(6.44)

Of course, these regularizations are different from 𝑓 at a point 𝑥 only if 𝑓 is not right- or left-continuous at 𝑥, respectively. In what follows, we will restrict ourselves to the right regularization (6.43). As we have seen⁶ in Proposition 4.28, the right regu­ larization (6.43) of a 𝐵𝑉-function is also a 𝐵𝑉-function and satisfies ‖𝑓#‖𝐵𝑉 ≤ ‖𝑓‖𝐵𝑉 ,

(6.45)

where the inequality may be strict. The following remarkable result was proved by Matkowski and Miś in [207], see also [226]. Theorem 6.19. If ℎ has the form (6.32) with 𝛼, 𝛽 ∈ 𝐵𝑉([𝑎, 𝑏]), the corresponding operator (6.1) satisfies a Lipschitz condition of type (6.30) in the space 𝐵𝑉([𝑎, 𝑏]) with norm (1.16). Conversely, suppose that the operator (6.1) maps the space 𝐵𝑉([𝑎, 𝑏]) with norm (1.16) into itself and satisfies a Lipschitz condition of type (6.30). Then the following is true. (a) The function ℎ(𝑥, ⋅) satisfies the Lipschitz condition (6.35) with 𝑘 = 𝐾. (b) The right regularization (6.43) of ℎ(⋅, 𝑢) has the form ℎ#(𝑥, 𝑢) = 𝛼(𝑥) + 𝛽(𝑥)𝑢

(𝑎 ≤ 𝑥 ≤ 𝑏, 𝑢 ∈ ℝ)

(6.46)

for some functions 𝛼, 𝛽 ∈ 𝐵𝑉([𝑎, 𝑏]). An analogous statement is true for the left regularization (6.44). Proof. The first statement follows from the fact that 𝐵𝑉([𝑎, 𝑏]) is an algebra, while (a) follows from Proposition 6.16 and the equality 𝑐(𝐵𝑉, 𝐵) = ‖𝑓1 ‖𝐵𝑉 = 1. The nontrivial statement is of course (b) for the converse. However, assertion (b) follows from a more general result (Theorem 6.29) which we are going to prove in the next section. In view of Theorem 6.19, it seems reasonable to introduce a weaker form of Defini­ tion 5.42. Definition 6.20. We say that a pair (𝑋, 𝑌) of two normed spaces (𝑋, ‖ ⋅ ‖𝑋 ) and (𝑌, ‖ ⋅ ‖𝑌 ) has the weak Matkowski property if whenever the operator (6.1) maps the space 𝑋 into the space 𝑌 and satisfies (6.30), the corresponding right regularization (6.43) of ℎ(⋅, 𝑢) must have the form (6.46). In case 𝑋 = 𝑌, we simply say that 𝑋 has the weak Matkowski property. ◼

6 To be precise, the regularization considered in Proposition 4.28 is different since we subtract 𝑓(𝑎) everywhere in (4.63), and thus “nail down” 𝑓# to be zero in 𝑎. However, the estimate (6.45) is of course also true here since functions which differ by a constant have the same total variation.

6.2 Lipschitz continuity

|

405

Of course, in the autonomous case of the composition operator (5.1), there is no dif­ ference between the Matkowski property and the weak Matkowski property. Theo­ rem 6.19 (b) states that the Banach space (𝐵𝑉([𝑎, 𝑏]), ‖ ⋅ ‖𝐵𝑉 ) has the weak Matkowski property. The following example which is a slight modification of an example given in [207] shows that this space does not have the Matkowski property in the sense of Definition 5.42. Therefore, in the nonautonomous case of the superposition operator (6.1), these notions are different. Example 6.21. Let {𝑟0 , 𝑟1 , 𝑟2 , . . .} be an enumeration of all rational numbers in [0, 1] (𝑟0 := 0), and let ℎ0 : ℝ → ℝ be any function satisfying ℎ0 (0) = 0 and |ℎ0 (𝑢) − ℎ0 (𝑣)| ≤ 𝐿|𝑢 − 𝑣|. We define ℎ : [0, 1] × ℝ → ℝ by { ℎ0 (𝑢) 𝑘 ℎ(𝑥, 𝑢) := { 2 0 {

if 𝑥 = 𝑟𝑘 , otherwise .

(6.47)

For any partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]) and 𝑓 ∈ 𝐵𝑉([0, 1]), we have 𝑚

∞

∑ |𝑆ℎ 𝑓(𝑡𝑗 ) − 𝑆ℎ 𝑓(𝑡𝑗−1 )| ≤ 2 ∑ |ℎ(𝑟𝑘 , 𝑓(𝑟𝑘 ))|

𝑗=1

𝑘=0 ∞

|ℎ (𝑓(𝑟 ))| = 2 ∑ 0 𝑘 𝑘 ≤ 4𝐿‖𝑓‖∞ , 2 𝑘=0

(6.48)

which shows that 𝑆ℎ maps the space 𝐵𝑉([0, 1]) into itself and is bounded. Furthermore, for 𝑓, 𝑔 ∈ 𝐵𝑉([0, 1]) and 𝑃 ∈ P([0, 1]), as above, we obtain the estimate Var(𝑆ℎ 𝑓 − 𝑆ℎ 𝑔, 𝑃; [0, 1]) 𝑚

= ∑ |𝑆ℎ 𝑓(𝑡𝑗 ) − 𝑆ℎ 𝑔(𝑡𝑗 ) − 𝑆ℎ 𝑓(𝑡𝑗−1 ) + 𝑆ℎ 𝑔(𝑡𝑗−1 )| 𝑗=1

𝑚

∞

≤ 2 ∑ |ℎ(𝑡𝑗 , 𝑓(𝑡𝑗 )) − ℎ(𝑡𝑗 , 𝑔(𝑡𝑗 ))| ≤ 2 ∑ |ℎ(𝑟𝑘 , 𝑓(𝑟𝑘 )) − ℎ(𝑟𝑘 , 𝑔(𝑟𝑘 ))| 𝑗=1

𝑘=0

∞

∞

|ℎ0 (𝑓(𝑟𝑘 )) − ℎ0 (𝑔(𝑟𝑘 ))| |𝑓(𝑟𝑘 ) − 𝑔(𝑟𝑘 ))| ≤ 2𝐿 ∑ 𝑘 2 2𝑘 𝑘=0 𝑘=0

≤2∑

∞

|𝑓(𝑟𝑘 ) − 𝑔(𝑟𝑘 ))| ≤ 2𝐿‖𝑓 − 𝑔‖𝐵𝑉 . 2𝑘 𝑘=1

= 2𝐿|𝑓(0) − 𝑔(0)| + 2𝐿 ∑

This together with the trivial estimate |𝑆ℎ 𝑓(0) − 𝑆ℎ 𝑔(0)| ≤ 𝐿|𝑓(0) − 𝑔(0)| shows that 𝑆ℎ satisfies the global Lipschitz condition (6.30) with 𝐾 = 2𝐿, although ℎ is not of the form (6.32). ♥ It is not hard to see that ℎ#(𝑥, 𝑢) = ℎ♭ (𝑥, 𝑢) ≡ 0 for the function ℎ in Example 6.21, in accordance with Theorem 6.19 (b). In the next section (Theorem 6.29), we will formulate a very general condition for a pair of Banach spaces under which this pair even has a stronger property than the weak Matkowski property.

406 | 6 Nonlinear superposition operators We do not reproduce the large summary at the end of Section 5.5 since conditions which are both necessary and sufficient for boundedness, continuity etc. are much more difficult to find (and, in fact, mostly unknown) in the nonautonomous case of the operator 𝑆ℎ . Therefore, the regions of terra incognita in such a summary would be vast: only in quite exceptional cases have we found explicit counterexamples for the operator (6.1), like the striking Examples 6.3 and 6.6. A small part of this “terra incognita” is covered by open problems stated in Section 6.5 below.

6.3 Uniform boundedness and continuity In the last section, we have encountered several spaces with the Matkowski property (respectively, the weak Matkowski property), meaning that a global Lipschitz condi­ tion on 𝑆ℎ implies that the generating function ℎ (respectively, its regularization ℎ#) is affine. We point out, however, that in some spaces, the degeneracy described by the Matkowski property also happens under the weaker assumption that 𝑆ℎ is uniformly bounded or uniformly continuous.⁷ Later in this section, we will take a closer look at this phenomenon, and we will also consider several concepts of uniform boundedness of 𝐶ℎ and 𝑆ℎ . One such concept reads as follows [203]: Definition 6.22. Let 𝑋 and 𝑌 be normed spaces and 𝐴 : 𝑋 → 𝑌 a (usually, nonlinear) operator. If there exists an increasing function 𝛾 : [0, ∞) → [0, ∞) such that 𝛾(0) = 0 and ‖𝐴𝑓 − 𝐴𝑔‖ ≤ 𝛾(‖𝑓 − 𝑔‖) (𝑓, 𝑔 ∈ 𝑋) , (6.49) the operator 𝐴 is called uniformly bounded.

◼

The following proposition establishes a link between uniform boundedness and uni­ form continuity. Proposition 6.23. Every operator which is uniformly continuous (for example, Lipschitz continuous) is also uniformly bounded. Conversely, if the function 𝛾 in (6.49) is continu­ ous at 0, then the uniform boundedness of 𝐴 implies its uniform continuity. Proof. Suppose that 𝐴 : 𝑋 → 𝑌 is uniformly continuous. This means that, given 𝜀 > 0, we find 𝛿 > 0 such that ‖𝑓 − 𝑔‖ ≤ 𝛿 implies ‖𝐴𝑓 − 𝐴𝑔‖ ≤ 𝜀 for 𝑓, 𝑔 ∈ 𝑋. Then putting 𝛾(𝑡) := sup {‖𝐴𝑓 − 𝐴𝑔‖ : ‖𝑓 − 𝑔‖ ≤ 𝑡}

(𝑡 ≥ 0) ,

(6.50)

7 In the autonomous case of the operator 𝐶ℎ , a typical result of this type is Corollary 5.46. In the nonau­ tonomous case, this was first observed and studied in [200].

6.3 Uniform boundedness and continuity

| 407

the function 𝛾 : [0, ∞) → [0, ∞) is well-defined and finite. Moreover, 𝛾 is obviously monotonically increasing. So, putting 𝑡 := ‖𝑓 − 𝑔‖ in (6.50), we obtain 𝛾(‖𝑓 − 𝑔‖) ≥ ‖𝐴𝑓 − 𝐴𝑔‖ as claimed. Conversely, suppose that (6.49) holds for some increasing function 𝛾 which is con­ tinuous at 0. Given 𝜀 > 0, choose 𝛿 > 0 such that 0 ≤ 𝑡 ≤ 𝛿 implies 0 ≤ 𝛾(𝑡) ≤ 𝜀. Fix any 𝑓, 𝑔 ∈ 𝑋 satisfying 𝑡 := ‖𝑓 − 𝑔‖ ≤ 𝛿. Then ‖𝐴𝑓 − 𝐴𝑔‖ ≤ 𝛾(‖𝑓 − 𝑔‖) ≤ 𝜀 .

(6.51)

However, this precisely means that 𝐴 is uniformly continuous on 𝑋. Here is a simple example of an operator which is uniformly bounded, but not uni­ formly continuous. Example 6.24. Consider the autonomous composition operator (5.1) generated by the function ℎ(𝑢) = sin 𝑢 between 𝑋 = 𝐿 1 ([0, 1]) and 𝑌 = 𝐿 ∞ ([0, 1]). Taking 𝛾(0) = 0 and 𝛾(𝑡) ≡ 2 for 𝑡 > 0, we see that (6.49) is true for 𝐴 = 𝐶ℎ , and so 𝐶ℎ is uniformly bounded between 𝑋 and 𝑌 in the sense of Definition 6.22. However, for the sequence of functions 𝑓𝑛 = 𝜋2 𝜒[0,1/𝑛] , we have ‖𝑓𝑛 ‖𝐿 1 → 0

(𝑛 → ∞) ,

but ‖𝐶ℎ 𝑓𝑛 ‖𝐿 ∞ = ‖𝜒[0,1/𝑛] ‖𝐿 ∞ ≡ 1 .

(6.52) ♥

This shows that 𝐶ℎ is not continuous at 𝑓(𝑥) ≡ 0.

From Proposition 6.23, it follows that (6.49) cannot hold for the operator 𝐴 = 𝐶ℎ in Example 6.24 with any function 𝛾 which is continuous at 0. Now, we will study the case when 𝐴 is the composition operator (5.1) or the super­ position operator (6.1). We start with the following analogue of Proposition 6.16. Proposition 6.25. Suppose that the operator (6.1) maps a normed space 𝑋 into a normed space 𝑌 and satisfies the uniform boundedness condition (6.49). Assume that the space 𝑋 contains the constant functions, and the space 𝑌 is imbedded into the space of bounded functions. Then the function ℎ is uniformly bounded in the sense that ̃ − 𝑣|) |ℎ(𝑥, 𝑢) − ℎ(𝑥, 𝑣)| ≤ 𝛾(|𝑢

(𝑎 ≤ 𝑥 ≤ 𝑏, 𝑢, 𝑣 ∈ ℝ) ,

(6.53)

̃ = 0. where 𝛾̃ : [0, ∞) → [0, ∞) is increasing with 𝛾(0) Proof. From (6.49) and our hypothesis 𝑌 󳨅→ 𝐵([𝑎, 𝑏]), it follows that ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖∞ ≤ 𝐾𝑐(𝑌, 𝐵)𝛾(‖𝑓 − 𝑔‖𝑋 )

(𝑓, 𝑔 ∈ 𝑋) ,

where we use the same notation as in Proposition 6.16. Choosing 𝑓(𝑥) ≡ 𝑢 and 𝑔(𝑥) ≡ 𝑣 in this estimate yields |ℎ(𝑥, 𝑢) − ℎ(𝑥, 𝑣)| ≤ ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖∞ ≤ 𝐾𝑐(𝑌, 𝐵)𝛾(‖𝑓1 ‖𝑋 |𝑢 − 𝑣|) , ̃ := 𝐾𝑐(𝑌, 𝐵)𝛾(‖𝑓1 ‖𝑋 𝑡). and so (6.53) holds with 𝛾(𝑡)

408 | 6 Nonlinear superposition operators Table 6.5. The Matkowski property. uniform Matkowski property ⇓ uniform weak Matkowski property

⇒ ⇒

Matkowski property ⇓ weak Matkowski property

In connection with uniform boundedness, we introduce a new notion which is parallel to both Definition 5.42 and Definition 6.20. Definition 6.26. We say that a pair (𝑋, 𝑌) of two normed spaces (𝑋, ‖ ⋅ ‖) and (𝑌, ‖ ⋅ ‖) has the uniform Matkowski property if whenever the operator (5.1) [the operator (6.1), respectively] maps the space 𝑋 into the space 𝑌 and is uniformly bounded, the corre­ sponding function ℎ must have the form (5.68) [the form (5.69), respectively]. In case 𝑋 = 𝑌, we simply say that 𝑋 has the uniform Matkowski property. Similarly, we say that a pair (𝑋, 𝑌) of two normed spaces (𝑋, ‖ ⋅ ‖) and (𝑌, ‖ ⋅ ‖) has the uniform weak Matkowski property if whenever the operator (6.1) maps the space 𝑋 into the space 𝑌 and is uniformly bounded, the corresponding right regularization (6.43) of ℎ must have the form (6.46). In case 𝑋 = 𝑌, we simply say that 𝑋 has the uniform weak Matkowski property. ◼ Putting 𝛾(𝑡) := 𝐾𝑡 in (6.49), we see that the uniform (weak) Matkowski property implies the (weak) Matkowski property. So, we have the following hierarchy between these properties. Now, we state two sufficient conditions on a pair of spaces (𝑋, 𝑌) to have the re­ spectively uniform weak Matkowski property. By 𝑃𝐿([𝑎, 𝑏]), we denote the set of all continuous piecewise linear functions, see Exercise 3.66. The following result is simi­ lar to Theorem 6.17. Theorem 6.27. Let 𝑋 and 𝑌 be two function spaces over [𝑎, 𝑏]. Assume that 𝑋 contains the space 𝑃𝐿([𝑎, 𝑏]), and 𝑌 is imbedded into the space 𝐴𝐶([𝑎, 𝑏]) with norm (3.42). Then (𝑋, 𝑌) has the uniform Matkowski property. Proof. We take again [𝑎, 𝑏] = [0, 1]. Recall that the norm (3.42) on 𝐴𝐶([0, 1]) is given by 1

‖𝑓‖𝐴𝐶 = ‖𝑓‖𝐵𝑉 = |𝑓(0)| + Var(𝑓; [0, 1]) = |𝑓(0)| + ∫ |𝑓󸀠 (𝑡)| 𝑑𝑡 . 0

Suppose that the superposition operator (6.1) maps 𝑋 into 𝑌 and satisfies (6.49) for some function 𝛾 : [0, ∞) → [0, ∞). From our hypothesis 𝑌 󳨅→ 𝐴𝐶([0, 1]) and the monotonicity of 𝛾, it follows that 1

∫ |(𝑆ℎ 𝑓)󸀠 (𝑡) − (𝑆ℎ 𝑔)󸀠 (𝑡)| 𝑑𝑡 ≤ ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖𝐴𝐶 ≤ 𝑐(𝑌, 𝐴𝐶)𝛾(‖𝑓 − 𝑔‖𝐴𝐶 ) 0

(6.54)

6.3 Uniform boundedness and continuity

| 409

for all 𝑓, 𝑔 ∈ 𝑋, where 𝑐(𝑌, 𝐴𝐶) denotes the imbedding constant (0.36). Fix numbers 𝑦1 , 𝑦2 , 𝑧1 , 𝑧2 ∈ ℝ and 𝑥1 , 𝑥2 , . . . , 𝑥2𝑛 ∈ (0, 1) such that 0 < 𝑥1 < 𝑥2 < . . . < 𝑥2𝑛 < 1 . Consider the piecewise linear functions 𝑓, 𝑔 : [0, 1] → ℝ defined by 𝑦1 { { { 𝑓(𝑥) := {𝑦2 { { {linear

for 𝑥 = 0 or 𝑥 ∈ {𝑥1 , 𝑥3 , . . . , 𝑥2𝑛−1 } ,

𝑧1 { { { 𝑔(𝑥) := {𝑧2 { { {linear

for 𝑥 = 0 or 𝑥 ∈ {𝑥1 , 𝑥3 , . . . , 𝑥2𝑛−1 } ,

for 𝑥 = 1 or 𝑥 ∈ {𝑥2 , 𝑥4 , . . . , 𝑥2𝑛 } , otherwise ,

and similarly

for 𝑥 = 1 or 𝑥 ∈ {𝑥2 , 𝑥4 , . . . , 𝑥2𝑛 } , otherwise .

So, 𝑓 is the unique piecewise linear functions whose graph is determined by the vertices (0, 𝑦1 ) , (𝑥1 , 𝑦1 ) ,

(𝑥2 , 𝑦2 ) ,

. . . , (𝑥2𝑘−1 , 𝑦1 ) , (𝑥2𝑘 , 𝑦2 ) ,

. . . , (𝑥2𝑛 , 𝑦2 ) ,

(1, 𝑦2 ) ,

and analogously for 𝑔. Thus, 𝑓(𝑥𝑘) = 𝑦1 if and only if 𝑓(𝑥𝑘+1 ) = 𝑦2 for 𝑘 = 1, 2, . . . , 2𝑛−1, and similarly for 𝑔 with 𝑦1 replaced by 𝑧1 and 𝑦2 replaced by 𝑧2 . Since 𝑓 and 𝑔 are constant on the intervals [0, 𝑥1 ] and [𝑥2𝑛 , 1], and affine on each of the other intervals [𝑥𝑘 , 𝑥𝑘+1 ], by definition of the norm (3.42), we have 2𝑛−1

𝑥𝑘+1

‖𝑓 − 𝑔‖AC = |𝑦1 − 𝑧1 | + ∑ ∫ |𝑓󸀠 (𝑡) − 𝑔󸀠 (𝑡)| 𝑑𝑡 𝑘=1 𝑥𝑘 2𝑛−1

󵄨 󵄨 󵄨 󵄨 = |𝑦1 − 𝑧1 | + ∑ 󵄨󵄨󵄨𝑦1 − 𝑧1 − 𝑦2 + 𝑧2 󵄨󵄨󵄨 = |𝑦1 − 𝑧1 | + (2𝑛 − 1) 󵄨󵄨󵄨𝑦1 − 𝑧1 − 𝑦2 + 𝑧2 󵄨󵄨󵄨 . 𝑘=1

On the other hand, 1

∫ |(𝑆ℎ 𝑓)󸀠 (𝑡) − (𝑆ℎ 𝑔)󸀠 (𝑡)| 𝑑𝑡 0 𝑥

2𝑛−1 𝑘+1 󵄨󵄨 󵄨󵄨 󵄨 󵄨𝑑 = ∑ ∫ 󵄨󵄨󵄨 [ℎ(𝑡, 𝑓(𝑡)) − ℎ(𝑡, 𝑔(𝑡))]󵄨󵄨󵄨 𝑑𝑡 󵄨󵄨 󵄨 𝑑𝑡 󵄨 𝑘=1 𝑥 𝑘

󵄨 𝑥𝑘+1 󵄨󵄨 󵄨󵄨 𝑑 󵄨󵄨 󵄨 [ℎ(𝑡, 𝑓(𝑡)) − ℎ(𝑡, 𝑔(𝑡))] 𝑑𝑡󵄨󵄨󵄨 ≥ ∑ 󵄨󵄨 ∫ 󵄨 󵄨󵄨 𝑑𝑡 𝑘=1 󵄨󵄨󵄨 𝑥𝑘 󵄨󵄨 2𝑛−1 󵄨󵄨󵄨

2𝑛−1

󵄨 󵄨 = ∑ 󵄨󵄨󵄨ℎ(𝑥𝑘+1 , 𝑓(𝑥𝑘+1 )) − ℎ(𝑥𝑘+1 , 𝑔(𝑥𝑘+1 )) − ℎ(𝑥𝑘 , 𝑓(𝑥𝑘 )) + ℎ(𝑥𝑘 , 𝑔(𝑥𝑘 ))󵄨󵄨󵄨 𝑘=1

2𝑛−1

󵄨 󵄨 = ∑ 󵄨󵄨󵄨ℎ(𝑥𝑘+1 , 𝑦1 ) − ℎ(𝑥𝑘+1 , 𝑧1 ) − ℎ(𝑥𝑘 , 𝑦2 ) + ℎ(𝑥𝑘 , 𝑧2 )󵄨󵄨󵄨 . 𝑘=1

410 | 6 Nonlinear superposition operators Inserting this into (6.54) yields 2𝑛−1

󵄨 󵄨 ∑ 󵄨󵄨󵄨ℎ(𝑥𝑘+1 , 𝑦1 ) − ℎ(𝑥𝑘+1 , 𝑧1 ) − ℎ(𝑥𝑘 , 𝑦2 ) + ℎ(𝑥𝑘 , 𝑧2 )󵄨󵄨󵄨

𝑘=1

󵄨 󵄨 ≤ 𝑐(𝑌, 𝐴𝐶)𝛾(|𝑦1 − 𝑧1 | + (2𝑛 − 1) 󵄨󵄨󵄨𝑦1 − 𝑧1 − 𝑦2 + 𝑧2 󵄨󵄨󵄨) . Fixing 𝑥 ∈ [0, 1] and letting 𝑥𝑘 tend to 𝑥 for all 𝑘 ∈ {1, 2, . . . , 2𝑛} gives 󵄨 󵄨 (2𝑛 − 1) 󵄨󵄨󵄨ℎ(𝑥, 𝑦1 ) − ℎ(𝑥, 𝑧1 ) − ℎ(𝑥, 𝑦2 ) + ℎ(𝑥, 𝑧2 )󵄨󵄨󵄨 󵄨 󵄨 ≤ 𝑐(𝑌, 𝐴𝐶)𝛾(|𝑦1 − 𝑧1 | + (2𝑛 − 1) 󵄨󵄨󵄨𝑦1 − 𝑧1 − 𝑦2 + 𝑧2 󵄨󵄨󵄨) .

(6.55)

Taking now arbitrary different points 𝑢, 𝑣 ∈ ℝ and making in (6.55) the special choice 𝑦1 :=

𝑢+𝑣 , 2

𝑦2 := 𝑢,

𝑧1 := 𝑣,

𝑧2 :=

𝑢+𝑣 , 2

we obtain 󵄨󵄨 𝑢 + 𝑣 󵄨󵄨󵄨 |𝑢 + 𝑣| 𝑢+𝑣 ) − ℎ(𝑥, 𝑢) − ℎ(𝑥, 𝑣) + ℎ (𝑥, )󵄨󵄨 ≤ 𝑐(𝑌, AC)𝛾 ( ). (2𝑛 − 1) 󵄨󵄨󵄨󵄨ℎ (𝑥, 2 2 󵄨󵄨 2 󵄨 Since 𝑛 ∈ ℕ is arbitrary, we conclude that necessarily 2ℎ (𝑥,

𝑢+𝑣 ) = ℎ(𝑥, 𝑣) + ℎ(𝑥, 𝑢) . 2

This means that the function ℎ(𝑥, ⋅) satisfies the Jensen functional equation, and therefore ℎ(𝑥, 𝑢) = 𝛼(𝑥) + 𝛽(𝑥)𝑢 (𝑎 ≤ 𝑥 ≤ 𝑏, 𝑢 ∈ ℝ) for some functions 𝛼, 𝛽 : [𝑎, 𝑏] → ℝ as claimed. The assertion 𝛼, 𝛽 ∈ 𝑌 follows from the fact that 𝛼(𝑥) = ℎ(𝑥, 0) and 𝛽(𝑥) = ℎ(𝑥, 1) − ℎ(𝑥, 0). By analyzing the hypothesis 𝑃𝐿 󳨅→ 𝑋 = 𝑌 󳨅→ 𝐴𝐶 in Theorem 6.27, we obtain the following Corollary 6.28. The spaces 𝐿𝑖𝑝, 𝐵𝑉1 , 𝑊𝐵𝑉1𝑝 , and 𝐴𝐶 have the uniform Matkowski prop­ erty. Unfortunately, the condition 𝑃𝐿 ⊆ 𝑋 in Theorem 6.27 excludes all spaces 𝑋 ⊆ 𝐶1 like 𝑅𝐵𝑉1𝑝 for 𝑝 > 1 or 𝐿𝑖𝑝1𝛼 for 𝛼 ≤ 1. On the other hand, the requirement 𝑌 ⊆ 𝐴𝐶 in Theorem 6.27 is less restrictive than the requirement 𝑌 ⊆ 𝐿𝑖𝑝 in Theorem 6.17. In particular, Theorem 6.27 covers the main result from [124] which states that 𝐴𝐶 has the uniform Matkowski property, while Theorem 6.17 does not. Now, we establish a sufficient condition under which a pair (𝑋, 𝑌) has the uniform weak Matkowski property. To this end, we will use the space 𝛷𝐵𝑉([𝑎, 𝑏]) of functions of bounded Schramm variation introduced in Definition 2.42. Recall that here 𝛷 = (𝜙𝑛 )𝑛 is a sequence of Young functions 𝜙𝑛 : [0, ∞) → [0, ∞) such that ∞

∑ 𝜙𝑘 (𝑥) = ∞

𝑘=1

(𝑥 > 0) .

(6.56)

6.3 Uniform boundedness and continuity

| 411

For special choices of (𝜙𝑛 )𝑛 , the space 𝛷𝐵𝑉 contains many of the spaces of func­ tions of generalized bounded variation we have studied so far, see Proposition 2.43. Theorem 6.29. Let 𝑋 and 𝑌 be two function spaces over [𝑎, 𝑏]. Assume that the space 𝑃𝑛 ([𝑎, 𝑏]) of polynomials of degree ≤ 𝑛, equipped with the norm of 𝑋, is imbedded into 𝑋 for each 𝑛 ∈ ℕ, and 𝑌 is imbedded into some space 𝛷𝐵𝑉([𝑎, 𝑏]) of functions of bounded Schramm variation with norm (2.75). Then (𝑋, 𝑌) has the uniform weak Matkowski prop­ erty. Proof. Given function spaces 𝑋 and 𝑌 with the indicated properties, suppose that the superposition operator (6.1) maps 𝑋 into 𝑌 and satisfies condition (6.49). From our hypotheses 𝑃𝑛 ([𝑎, 𝑏]) 󳨅→ 𝑋 and 𝑌 󳨅→ 𝛷𝐵𝑉([𝑎, 𝑏]) and the monotonicity of 𝛾 in (6.49), it follows that ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖𝛷𝐵𝑉 ≤ 𝑐(𝑌, 𝛷𝐵𝑉)𝛾(‖𝑓 − 𝑔‖𝑋 )

(𝑓, 𝑔 ∈ 𝑃𝑛 ([𝑎, 𝑏])) ,

(6.57)

where 𝑐(𝑌, 𝛷𝐵𝑉) denotes the imbedding constant (0.36). Let 𝑎 ≤ 𝑡 < 𝑠 ≤ 𝑏, and let 𝑃𝑚 := {𝑡0 , 𝑡1 , . . . , 𝑡2𝑚 } ∈ P([𝑡, 𝑠]) be the equidistant partition defined by 𝑡𝑗 − 𝑡𝑗−1 =

𝑠−𝑡 2𝑚

(𝑗 = 1, 2, . . . , 2𝑚) .

(6.58)

Given 𝑢, 𝑣 ∈ ℝ with 𝑢 ≠ 𝑣, let 𝑓 : [𝑎, 𝑏] → ℝ be a polynomial satisfying 𝑓(𝑡2𝑗 ) = 𝑣

(𝑗 = 0, 1, . . . , 𝑚),

𝑓(𝑡2𝑗−1 ) =

𝑢+𝑣 2

(𝑗 = 1, 2, . . . , 𝑚) ,

(6.59)

(𝑗 = 1, 2, . . . , 𝑚) ,

(6.60)

and let 𝑔 : [𝑎, 𝑏] → ℝ be the polynomial 𝑔(𝑥) := 𝑓(𝑥) +

𝑢−𝑣 . 2

Then 𝑔(𝑡2𝑗 ) =

𝑢+𝑣 2

(𝑗 = 0, 1, . . . , 𝑚),

𝑔(𝑡2𝑗−1 ) = 𝑢

and the difference 𝑓 − 𝑔 trivially satisfies |𝑓(𝑥) − 𝑔(𝑥)| ≡

|𝑢 − 𝑣| 2

(𝑎 ≤ 𝑥 ≤ 𝑏) .

Consequently, substituting these functions 𝑓 and 𝑔 into (6.57) yields ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖𝛷𝐵𝑉 ≤ 𝑐(𝑌, 𝛷𝐵𝑉)𝛾(‖𝑓1 ‖𝑋 |𝑢 − 𝑣|/2) , and hence

󵄩󵄩 󵄩󵄩 𝑆ℎ 𝑓 − 𝑆ℎ 𝑔 󵄩󵄩 󵄩󵄩 ≤ 1, 󵄩󵄩 󵄩 󵄩󵄩 𝑐(𝑌, 𝛷𝐵𝑉)𝛾(‖𝑓1 ‖𝑋 |𝑢 − 𝑣|/2) 󵄩󵄩󵄩𝛷𝐵𝑉

where again 𝑓1 (𝑥) ≡ 1. From Proposition 2.44 (b), it follows that Var𝛷 (

󵄩󵄩 󵄩󵄩 𝑆ℎ 𝑓 − 𝑆ℎ 𝑔 𝑆ℎ 𝑓 − 𝑆ℎ 𝑔 󵄩 󵄩󵄩 ) ≤ 󵄩󵄩󵄩 ≤ 1. 󵄩 󵄩󵄩 𝑐(𝑌, 𝛷𝐵𝑉)𝛾(‖𝑓1 ‖𝑋 |𝑢 − 𝑣|/2) 󵄩󵄩󵄩𝛷𝐵𝑉 𝑐(𝑌, 𝛷𝐵𝑉)𝛾(‖𝑓1 ‖𝑋 |𝑢 − 𝑣|/2)

(6.61)

412 | 6 Nonlinear superposition operators Building on the definition of Var𝛷 (𝑓) and using (6.59), we therefore get 𝑚

∑ 𝜙𝑗 (

|ℎ(𝑡2𝑗 , 𝑓(𝑡2𝑗 )) − ℎ(𝑡2𝑗 , 𝑔(𝑡2𝑗 )) − ℎ(𝑡2𝑗−1 , 𝑓(𝑡2𝑗−1 )) + ℎ(𝑡2𝑗−1 , 𝑔(𝑡2𝑗−1 ))| 𝑐(𝑌, 𝛷𝐵𝑉)𝛾(‖𝑓1 ‖𝑋 |𝑢 − 𝑣|/2)

𝑗=1

)

󵄨 󵄨 𝑚 2 󵄨󵄨󵄨ℎ(𝑡2𝑗 , 𝑣) − ℎ(𝑡2𝑗 , 𝑢+𝑣 ) − ℎ(𝑡2𝑗−1 , 𝑢+𝑣 ) + ℎ(𝑡2𝑗−1 , 𝑢)󵄨󵄨󵄨󵄨 2 2 ) ≤ 1. = ∑ 𝜙𝑗 ( 󵄨 𝑐(𝑌, 𝛷𝐵𝑉)𝛾(‖𝑓1 ‖𝑋 |𝑢 − 𝑣|/2) 𝑗=1 Now, since the operator 𝑆ℎ maps the space of all polynomials into the space 𝛷𝐵𝑉([𝑎, 𝑏]), the function 𝑓(⋅, 𝑧) has unilateral limits, for all 𝑧 ∈ ℝ, in the interval [𝑎, 𝑏]. Therefore, we may pass to the right-hand limit 𝑠 → 𝑡+ in the last estimate and obtain⁸ 󵄨󵄨 # 󵄨 𝑚 󵄨󵄨ℎ (𝑡, 𝑣) − ℎ#(𝑡, 𝑢+𝑣 ) − ℎ#(𝑡, 𝑢+𝑣 ) + ℎ#(𝑡, 𝑢)󵄨󵄨󵄨󵄨 2 2 󵄨 )≤1 (6.62) ∑ 𝜙𝑗 ( 𝑐(𝑌, 𝛷𝐵𝑉)𝛾(‖𝑓1 ‖𝑋 |𝑢 − 𝑣|/2) 𝑗=1 since 𝑠 → 𝑡+ implies 𝑡𝑗 → 𝑡+ for any 𝑗, by (6.58). Now, passing to the limit 𝑚 → ∞ in (6.62), we obtain 󵄨󵄨󵄨ℎ#(𝑡, 𝑣) − ℎ#(𝑡, 𝑢+𝑣 ) − ℎ#(𝑡, 𝑢+𝑣 ) + ℎ#(𝑡, 𝑢)󵄨󵄨󵄨 ∞ 󵄨 󵄨󵄨 2 2 ) ≤ 1. ∑ 𝜙𝑗 ( 󵄨 𝑐(𝑌, 𝛷𝐵𝑉)𝛾(‖𝑓 ‖ |𝑢 − 𝑣|/2) 1 𝑋 𝑗=1 However, (6.56) implies that this is possible only if ℎ#(𝑡, 𝑣) − ℎ# (𝑡,

𝑢+𝑣 𝑢+𝑣 ) − ℎ# (𝑡, ) + ℎ#(𝑡, 𝑢) = 0 . 2 2

This means that the function ℎ# satisfies the Cauchy functional equation 2ℎ# (𝑡,

𝑢+𝑣 ) = ℎ#(𝑡, 𝑢) + ℎ#(𝑡, 𝑣) 2

(𝑎 ≤ 𝑡 ≤ 𝑏, 𝑢, 𝑣 ∈ ℝ) .

Since ℎ# is right-continuous, it follows that (6.46) holds for some functions 𝛼, 𝛽 : [𝑎, 𝑏] → ℝ as claimed. The assertion 𝛼, 𝛽 ∈ 𝑌 follows from the fact that 𝛼(𝑥) = ℎ#(𝑥, 0) and 𝛽(𝑥) = ℎ#(𝑥, 1) − ℎ#(𝑥, 0). Theorem 6.29 applies to several function spaces which we have discussed in Chapter 1 and Chapter 2. In fact, from Proposition 2.43, we immediately get the following Corollary 6.30. The spaces 𝐵𝑉, Λ𝐵𝑉, 𝑊𝐵𝑉𝑝 , 𝑊𝐵𝑉𝜙 , and 𝛷𝐵𝑉 have the uniform weak Matkowski property. Corollary 6.30 was proved by a direct construction by Matkowski and Miś for the space 𝐵𝑉 in [207], see Theorem 6.19, by Merentes and Rivas for the space 𝑊𝐵𝑉𝑝 in [224] and for the space 𝑊𝐵𝑉𝜙 in [226].

8 Letting 𝑠 → 𝑡+ means that we “squeeze” the partition 𝑃𝑚 towards the singleton {𝑡}. Since ℎ#(⋅, 𝑢) is right-continuous, in (6.62), we then get the first argument 𝑡 in ℎ# throughout.

6.3 Uniform boundedness and continuity

| 413

Example 6.21 and Corollary 6.30 show that the space 𝐵𝑉 has the uniform weak Matkowski property, but not the Matkowski property, so neither of the vertical arrows in Table 6.5 may be inverted. We do not know whether or not the horizontal arrows can be inverted either. Of course, Example 6.21 shows that the spaces mentioned in Corollary 6.30 have the Matkowski property since 𝐵𝑉 is a special case of the other spaces. In the following Tables 6.6 and 6.7, we summarize some imbedding conditions on 𝑋 and 𝑌 under which either the global Lipschitz condition (6.30) or the uniform boundedness condition (6.49) for the operator 𝑆ℎ implies a degeneracy for the corre­ sponding function ℎ. Afterwards, we collect some spaces which have the Matkowski property, weak Matkowski property, uniform Matkowski property, or uniform weak Matkowski property in Table 6.8. Table 6.6. Globally Lipschitz superposition operators. If 𝑋 satisfies ℝ 󳨅→ 𝑋 𝑃1 ([𝑎, 𝑏]) 󳨅→ 𝑋 𝑃𝑛 ([𝑎, 𝑏]) 󳨅→ 𝑋

and 𝑌 satisfies

then (6.30) implies

𝑌 󳨅→ 𝐵([𝑎, 𝑏]) 𝑌 󳨅→ 𝐿𝑖𝑝([𝑎, 𝑏]) 𝑌 󳨅→ 𝛷𝐵𝑉([𝑎, 𝑏])

ℎ(𝑥, ⋅) locally Lipschitz ℎ(𝑥, ⋅) affine ℎ#(𝑥, ⋅) affine

Table 6.7. Uniformly bounded superposition operators. If 𝑋 satisfies ℝ 󳨅→ 𝑋 𝑃𝐿([𝑎, 𝑏]) 󳨅→ 𝑋 𝑃𝑛 ([𝑎, 𝑏]) 󳨅→ 𝑋

and 𝑌 satisfies

then (6.49) implies

𝑌 󳨅→ 𝐵([𝑎, 𝑏]) 𝑌 󳨅→ 𝐴𝐶([𝑎, 𝑏]) 𝑌 󳨅→ 𝛷𝐵𝑉([𝑎, 𝑏])

ℎ(𝑥, ⋅) uniformly bounded ℎ(𝑥, ⋅) affine ℎ#(𝑥, ⋅) affine

Note that Table 6.8 not only contains results which we have proved before, but also some results from references which are not covered by Corollaries 6.18, 6.28, or 6.30. Specifically, the uniform Matkowski property of 𝑅𝐵𝑉𝑝 was proved in [27], of 𝑅𝐵𝑉𝜙 in [1], and for 𝐿𝑖𝑝𝛼 in [201]. Historically, 𝐶1 , 𝐿𝑖𝑝, and 𝐿𝑖𝑝𝛼 have been the first spaces for which the Matkowski property in the sense of Definition 5.42 has been discovered. The stronger result that they also have the uniform Matkowski property has been proved more recently in [200, 201]. There are many other spaces with the (uniform or simple) Matkowski property. Let us just remark that in [32] it was shown quite recently that the space 𝜅𝐵𝑉 of functions of bounded Korenblum variation (Section 2.5) has the Matkowski property. This means that the condition Var𝜅 (𝑆ℎ 𝑓 − 𝑆ℎ 𝑔; [0, 1]) ≤ 𝐾 Var𝜅 (𝑓 − 𝑔; [0, 1])

(𝑓, 𝑔 ∈ 𝜅𝐵𝑉([0, 1])) ,

where Var𝜅 (𝑓 − 𝑔; [0, 1]) denotes the Korenblum variation (2.106), is satisfied only for affine functions (6.32).

414 | 6 Nonlinear superposition operators Table 6.8. Spaces with the Matkowski property. function space

Matkowski property

weak M. property

uniform M. property

uniform weak M. property

𝐶([𝑎, 𝑏]) 𝐶1 ([𝑎, 𝑏]) 𝐿𝑖𝑝([𝑎, 𝑏]) 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) 𝐴𝐶([𝑎, 𝑏]) 𝐵𝑉([𝑎, 𝑏]) 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) Λ𝐵𝑉([𝑎, 𝑏]) 𝛷𝐵𝑉([𝑎, 𝑏]) 𝐿𝑖𝑝1 ([𝑎, 𝑏]) 𝐿𝑖𝑝1𝛼 ([𝑎, 𝑏]) 𝐴𝐶1 ([𝑎, 𝑏]) 𝐵𝑉1 ([𝑎, 𝑏]) 𝑊𝐵𝑉1𝑝 ([𝑎, 𝑏]) 𝑅𝐵𝑉1𝑝 ([𝑎, 𝑏])

no (Th. 6.14) yes (Cor. 6.18) yes (Cor. 6.18) yes [193] yes (Cor. 6.28) no (Ex. 6.21) no (Ex. 6.21) yes [27] no (Ex. 6.21) no (Ex. 6.21) yes (Cor. 6.18) yes (Cor. 6.18) yes (Cor. 6.18) yes (Cor. 6.18) yes (Cor. 6.18) yes (Cor. 6.18)

– – – – – yes (Th. 6.19) yes (Cor. 6.30) – yes (Cor. 6.30) yes (Cor. 6.30) – – – – – –

no (Th. 6.14) yes [200] yes (Cor. 6.28) yes [201] yes (Cor. 6.28) no (Ex. 6.21) no (Ex. 6.21) yes [27] no (Ex. 6.21) no (Ex. 6.21) yes [200] yes yes [200] yes (Cor. 6.28) yes (Cor. 6.28) yes [200]

– – – – – yes (Cor. 6.30) yes (Cor. 6.30) – yes (Cor. 6.30) yes (Cor. 6.30) – – – – – –

Following [124], we now state two conditions which are equivalent to uniform boundedness. Proposition 6.31. Let 𝑋 and 𝑌 be two normed spaces and 𝐴 : 𝑋 → 𝑌 be a bounded operator. Then the following three conditions are equivalent: (a) The operator 𝐴 is uniformly bounded. (b) There exist constants 𝑅, 𝑟 > 0 such that ‖𝐴𝑓 − 𝐴𝑔‖ ≤ 𝑅 for all 𝑓, 𝑔 ∈ 𝑋 satisfying ‖𝑓 − 𝑔‖ = 𝑟. (c) There exist constants 𝑅,̃ 𝑟 ̃ > 0 such that ‖𝐴𝑓 − 𝐴𝑔‖ ≤ 𝑅̃ for all 𝑓, 𝑔 ∈ 𝑋 satisfying ‖𝑓 − 𝑔‖ ≤ 𝑟.̃ Proof. Obviously, (a) implies (b) by taking 𝑅 := 𝛾(𝑟), where 𝛾 is the function occurring in (6.49). Suppose that (b) is true, and take 𝑟 ̃ := 2𝑟 and 𝑅̃ := 2𝑅. Given 𝑓, 𝑔 ∈ 𝑋 satisfy­ ing ‖𝑓 − 𝑔‖ ≤ 𝑟,̃ we may find an element ℎ ∈ 𝑋 such that ‖𝑓 − ℎ‖ = ‖ℎ − 𝑔‖ = 𝑟 because the two spheres {ℎ ∈ 𝑋 : ‖𝑓 − ℎ‖ = 𝑟} and {ℎ ∈ 𝑋 : ‖𝑔 − ℎ‖ = 𝑟} have a nonempty intersection. By (b), there exists 𝑅 > 0 such that ‖𝐴𝑓 − 𝐴ℎ‖ ≤ 𝑅 and ‖𝐴ℎ − 𝐴𝑔‖ ≤ 𝑅, and hence ‖𝐴𝑓 − 𝐴𝑔‖ ≤ ‖𝐴𝑓 − 𝐴ℎ‖ + ‖𝐴ℎ − 𝐴𝑔‖ ≤ 2𝑅 = 𝑅̃ , which shows that (c) holds. The fact that (c) implies (b) is trivial, so it remains to prove that (b) implies (a). Putting, under the hypothesis (b), ̃ := sup {‖𝐴𝑓 − 𝐴𝑔‖ : ‖𝑓 − 𝑔‖ = 𝑡} 𝛾(𝑡)

(𝑡 ≥ 0) ,

(6.63)

6.4 Functions of several variables

| 415

we see that the function 𝛾̃ : [0, ∞) → [0, ∞) is well-defined and finite. Moreover, ̃ = 0 and, given 𝑓, 𝑔 ∈ 𝑋 with ‖𝑓 − 𝑔‖ = 𝑡, by (6.63), we have 𝛾(0) ̃ = 𝛾(‖𝑓 ̃ ‖𝐴𝑓 − 𝐴𝑔‖ ≤ 𝛾(𝑡) − 𝑔‖) .

(6.64)

However, in contrast to (6.50), the function (6.63) is not necessarily increasing. This flaw may be removed by defining 𝛾 : [0, ∞) → [0, ∞) by ̃ : 0 ≤ 𝑠 ≤ 𝑡} 𝛾(𝑡) := sup {𝛾(𝑠)

(𝑡 ≥ 0) .

Then 𝛾 is certainly increasing, and the operator 𝐴 satisfies (6.49) with this modi­ fied function 𝛾. Our discussion shows that all three conditions in Proposition 6.31 are equivalent to the uniform boundedness of 𝐴 in the sense of Definition 6.22. However, another variant of this will be given in Section 6.5.

6.4 Functions of several variables Given a function 𝑓 : [𝑎, 𝑏] → ℝ, in (6.43), we have introduced and studied the right reg­ ularization 𝑓#and, in (6.44), the left regularization 𝑓♭ of 𝑓. In terms of these regulariza­ tions, we could formulate in Theorem 6.19 a necessary conditions for the superposition operator (6.1) to satisfy a global Lipschitz condition in 𝐵𝑉 and related spaces. We are now going to formulate a parallel result for functions of two variables, which means that we replace the interval [𝑎, 𝑏] by the rectangle [𝑎, 𝑏] × [𝑐, 𝑑]. Following Chistyakov [90], we consider (double) left regularizations here which generalize (6.44). Definition 6.32. Given 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ, we define a function 𝑓♭♭ : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ by {lim(𝑠,𝑡)→(𝑥−,𝑦−) 𝑓(𝑠, 𝑡) { { { { {lim(𝑠,𝑡)→(𝑥−,𝑐+) 𝑓(𝑠, 𝑡) ♭♭ 𝑓 (𝑥, 𝑦) := { {lim { (𝑠,𝑡)→(𝑎+,𝑦−) 𝑓(𝑠, 𝑡) { { { {lim(𝑠,𝑡)→(𝑎+,𝑐+) 𝑓(𝑠, 𝑡)

for 𝑎 < 𝑥 ≤ 𝑏 and 𝑐 < 𝑦 ≤ 𝑑 , for 𝑎 < 𝑥 ≤ 𝑏 and 𝑦 = 𝑐 , for 𝑥 = 𝑎 and 𝑐 < 𝑦 ≤ 𝑑 ,

(6.65)

for 𝑥 = 𝑎 and 𝑦 = 𝑐 .

In what follows, we call 𝑓♭♭ the left-left regularization of 𝑓.

◼

In Definition 6.32, we distinguish between arguments in the upper right half-open rectangle (𝑎, 𝑏] × (𝑐, 𝑑], the half-open bottom side (𝑎, 𝑏] × {𝑐}, the half-open left side {𝑎} × (𝑐, 𝑑], and the left-bottom corner point (𝑎, 𝑐). Of course, other distinctions are also possible which lead to other types of regularizations, see Exercise 6.8. Recall that a function 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ is called left-left continuous on (𝑎, 𝑏] × (𝑐, 𝑑] if 𝑓(𝑠, 𝑡) = 𝑓(𝑥, 𝑦) (𝑎 < 𝑥 ≤ 𝑏, 𝑐 < 𝑦 ≤ 𝑑) . (6.66) lim (𝑠,𝑡)→(𝑥−,𝑦−)

416 | 6 Nonlinear superposition operators By 𝐵𝑉♭♭ ([𝑎, 𝑏] × [𝑐, 𝑑]), we denote the set of all functions 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) which are left-left continuous on (𝑎, 𝑏] × (𝑐, 𝑑]. Proposition 6.33. From 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]), it follows that 𝑓♭♭ ∈ 𝐵𝑉♭♭ ([𝑎, 𝑏] × [𝑐, 𝑑]). Moreover, ‖𝑓♭♭ ‖𝐵𝑉 ≤ ‖𝑓‖𝐵𝑉 . (6.67) Proof. A comparison of (6.65) and (6.66) shows that the left-left regularization 𝑓♭♭ is always left-left continuous on (𝑎, 𝑏] × (𝑐, 𝑑]; so, we only have to show that 𝑓♭♭ is of bounded variation if 𝑓 is. Moreover, we modify the proof of (4.64) to prove (6.67). The finiteness of the one-dimensional variations (1.76) and (1.77) is a consequence of Proposition 4.28. To see that the two-dimensional variation (1.78) is also finite, we fix {𝑠0 , 𝑠1 , . . . , 𝑠𝑚 } ∈ P([𝑎, 𝑏]), {𝑡0 , 𝑡1 , . . . , 𝑡𝑛 } ∈ P([𝑐, 𝑑]), and 𝜀 > 0. By definition (6.65) of the left-left regularization, we may find elements 𝜎𝑖 ∈ (𝑠𝑖−1 , 𝑠𝑖 ) (𝑖 = 1, 2, . . . , 𝑚) and 𝜏𝑗 ∈ (𝑡𝑗−1 , 𝑡𝑗 ) (𝑗 = 1, 2, . . . , 𝑛) as well as 𝜎0 ∈ (𝑎, 𝑠1 ) and 𝜏0 ∈ (𝑐, 𝑡1 ) such that 󵄨 󵄨󵄨 ♭♭ 󵄨󵄨𝑓 (𝑠𝑖−1 , 𝑡𝑗−1 ) + 𝑓♭♭ (𝑠𝑖 , 𝑡𝑗 ) − 𝑓♭♭ (𝑠𝑖−1 , 𝑡𝑗 ) + 𝑓♭♭ (𝑠𝑖 , 𝑡𝑗−1 )󵄨󵄨󵄨 󵄨 󵄨 𝜀 󵄨 󵄨 . ≤ 󵄨󵄨󵄨󵄨𝑓(𝜎𝑖−1 , 𝜏𝑗−1 ) + 𝑓(𝜎𝑖 , 𝜏𝑗 ) − 𝑓(𝜎𝑖−1 , 𝜏𝑗 ) + 𝑓(𝜎𝑖 , 𝜏𝑗−1 )󵄨󵄨󵄨󵄨 + 𝑚𝑛 It follows that 𝑚 𝑛 󵄨 󵄨 ∑ ∑ 󵄨󵄨󵄨󵄨𝑓♭♭ (𝑠𝑖−1 , 𝑡𝑗−1 ) + 𝑓♭♭ (𝑠𝑖 , 𝑡𝑗 ) − 𝑓♭♭ (𝑠𝑖−1 , 𝑡𝑗 ) + 𝑓♭♭ (𝑠𝑖 , 𝑡𝑗−1 )󵄨󵄨󵄨󵄨 𝑖=1 𝑗=1

𝑚 𝑛 󵄨 󵄨 𝜀 ≤ ∑ ∑ 󵄨󵄨󵄨󵄨𝑓(𝜎𝑖−1 , 𝜏𝑗−1 ) + 𝑓(𝜎𝑖 , 𝜏𝑗 ) − 𝑓(𝜎𝑖−1 , 𝜏𝑗 ) + 𝑓(𝜎𝑖 , 𝜏𝑗−1 )󵄨󵄨󵄨󵄨 + 𝑚𝑛 𝑖=1 𝑗=1

≤ V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) + 𝜀 , where V2 (𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) denotes the variation (1.78). Since 𝜀 > 0 was arbitrary, we have shown that 𝑓 ∈ 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) implies 𝑓♭♭ ∈ 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) with (6.67). We now state a result for functions of two variables which is parallel to Theorem 6.19. First, we have to define nonautonomous superposition operators acting on functions of two variables. This is of course straightforward. Given a function ℎ : [𝑎, 𝑏] × [𝑐, 𝑑] × ℝ → ℝ and a space 𝑋 of functions 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ, we define the superposition operator 𝑆ℎ generated by ℎ as 𝑆ℎ 𝑓(𝑥, 𝑦) := ℎ(𝑥, 𝑦, 𝑓(𝑥, 𝑦))

(𝑎 ≤ 𝑥 ≤ 𝑏, 𝑐 ≤ 𝑦 ≤ 𝑑) .

(6.68)

Now, we formulate an analogue of Theorem 6.19 for the space 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]). Theorem 6.34. If ℎ has the form ℎ(𝑥, 𝑦, 𝑢) = 𝛼(𝑥, 𝑦) + 𝛽(𝑥, 𝑦)𝑢

(𝑎 ≤ 𝑥 ≤ 𝑏, 𝑐 ≤ 𝑦 ≤ 𝑑, 𝑢 ∈ ℝ)

(6.69)

6.4 Functions of several variables

| 417

with 𝛼, 𝛽 ∈ 𝐵𝑉([𝑎, 𝑏]× [𝑐, 𝑑]), the corresponding operator (6.68) satisfies a Lipschitz con­ dition of type (6.30) in the space 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) with norm (1.89). Conversely, suppose that the operator (6.68) maps the space 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) with norm (1.89) into itself and satisfies a Lipschitz condition of type (6.30). Then the following is true. (a) The function ℎ(𝑥, 𝑦, ⋅) satisfies the Lipschitz condition |ℎ(𝑥, 𝑦, 𝑢) − ℎ(𝑥, 𝑦, 𝑣)| ≤ 2𝐾|𝑢 − 𝑣| (𝑎 ≤ 𝑥 ≤ 𝑏, 𝑐 ≤ 𝑦 ≤ 𝑑, 𝑢, 𝑣 ∈ ℝ) . (b) The left-left regularization (6.65) of ℎ(⋅, ⋅, 𝑢) has the form ℎ♭♭ (𝑥, 𝑦, 𝑢) = 𝛼(𝑥, 𝑦) + 𝛽(𝑥, 𝑦)𝑢

(𝑎 ≤ 𝑥 ≤ 𝑏, 𝑐 ≤ 𝑦 ≤ 𝑑, 𝑢 ∈ ℝ)

(6.70)

for some functions 𝛼, 𝛽 ∈ 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]). We do not carry out the proof in detail since it is, in large part, parallel to that for func­ tions of one variable. The first statement follows easily from the fact that 𝐵𝑉([𝑎, 𝑏] × [𝑐, 𝑑]) is an algebra (Proposition 1.44). For the proof of (a), one has to consider suitable generalizations of the bridge function (6.37) and to distinguish the four cases arising in the definition (6.65) of the left-left regularization. Finally, statement (b) is proved, with some technical modifications, in the same way as we have done this in Theorem 6.29 for functions of one variable. We drop the details and refer to the paper [90]. The following example which is parallel to Example 6.21 and involves a function ℎ satisfying (6.70), but not (6.69), is also taken from [90]. Example 6.35. Let {𝑟0 , 𝑟1 , 𝑟2 , . . .} be an enumeration of all rational numbers in [0, 1] (𝑟0 := 0), and let ℎ0 : ℝ → ℝ be any function satisfying ℎ0 (0) = 0 and |ℎ0 (𝑢) − ℎ0 (𝑣)| ≤ 𝐿|𝑢 − 𝑣|. We define ℎ : [0, 1] × [0, 1] × ℝ → ℝ by (𝑢) { ℎ0𝑘+𝑙 ℎ(𝑥, 𝑦, 𝑢) := { 2 0 {

if 𝑥 = 𝑟𝑘 and 𝑦 = 𝑟𝑙 , otherwise .

We show that Var(𝑆ℎ 𝑓; [0, 1]×[0, 1]) is finite for every function 𝑓 ∈ 𝐵𝑉([0, 1]×[0, 1]), where Var(𝑓; [𝑎, 𝑏] × [𝑐, 𝑑]) is given by (1.82). For any two partitions 𝑃 = {𝑠0 , 𝑠1 , . . . , 𝑠𝑚 } ∈ P([0, 1]) and 𝑄 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑛 } ∈ P([0, 1]), we have V2 (𝑆ℎ 𝑓, 𝑃 × 𝑄; [0, 1] × [0, 1]) 𝑛

𝑚

= ∑ ∑ |𝑆ℎ 𝑓(𝑠𝑖−1 , 𝑡𝑗−1 ) − 𝑆ℎ 𝑓(𝑠𝑖 , 𝑡𝑗 ) − 𝑆ℎ 𝑓(𝑠𝑖−1 , 𝑡𝑗 ) + 𝑆ℎ 𝑓(𝑠𝑖 , 𝑡𝑗−1 )| 𝑖=1 𝑗=1 𝑚

𝑛

∞ ∞

≤ 4 ∑ ∑ |ℎ(𝑠𝑖 , 𝑡𝑗 , 𝑓(𝑠𝑖 , 𝑡𝑗 ))| ≤ 4 ∑ ∑ |ℎ(𝑟𝑘 , 𝑟𝑙 , 𝑓(𝑟𝑘 , 𝑟𝑙 ))| 𝑖=0 𝑗=0

𝑘=1 𝑙=1

∞ ∞

∞ ∞ |ℎ (𝑓(𝑟 , 𝑟 ))| 1 = 4 ∑ ∑ 0 𝑘+𝑙𝑘 𝑙 ≤ 4𝐿‖𝑓‖∞ ∑ ∑ 𝑘+𝑙 ≤ 4𝐿‖𝑓‖∞ , 2 𝑘=1 𝑙=1 𝑘=1 𝑙=1 2

which shows that 𝑉(𝑆ℎ 𝑓; [0, 1] × [0, 1]) ≤ 4𝐿‖𝑓‖∞ .

418 | 6 Nonlinear superposition operators The one-dimensional variation Var(𝑆ℎ 𝑓(⋅, 0), 𝑃; [0, 1]) occurring in (1.76) may be es­ timated by 𝑚

Var(𝑆ℎ 𝑓(⋅, 0), 𝑃; [0, 1]) = ∑ |𝑆ℎ 𝑓(𝑠𝑖 , 0) − 𝑆ℎ 𝑓(𝑠𝑖−1 , 0)| 𝑖=1 𝑚

≤ 2 ∑ |ℎ(𝑠𝑖 , 0, 𝑓(𝑠𝑖 , 0))| 𝑖=1 ∞

∞

1 = 2𝐿‖𝑓‖∞ , 𝑘 𝑘=1 2

≤ 2 ∑ |ℎ(𝑟𝑘 , 0, 𝑓(𝑟𝑘 , 0))| ≤ 2𝐿‖𝑓‖∞ ∑ 𝑘=1

and similarly for the one-dimensional variation Var(𝑆ℎ 𝑓(0, ⋅), 𝑄; [0, 1]) occurring in (1.77). This shows that Var(𝑆ℎ 𝑓; [0, 1] × [0, 1]) ≤ 8𝐿‖𝑓‖∞ , and so the superposition operator (6.68) generated by ℎ maps the space 𝐵𝑉([0, 1] × [0, 1]) into itself and is bounded. Now, we prove that the operator 𝑆ℎ satisfies a global Lipschitz condition (6.30). Fix 𝑓, 𝑔 ∈ 𝐵𝑉([0, 1] × [0, 1]) and 𝑃, 𝑄 ∈ P([0, 1]) as above. Then using the shortcut 𝛥(𝑥, 𝑦) := (𝑆ℎ 𝑓 − 𝑆ℎ 𝑔)(𝑥, 𝑦) = ℎ(𝑥, 𝑦, 𝑓(𝑥, 𝑦)) − ℎ(𝑥, 𝑦, 𝑔(𝑥, 𝑦)), we obtain V2 (𝑆ℎ 𝑓 − 𝑆ℎ 𝑔, 𝑃 × 𝑄; [0, 1] × [0, 1]) 𝑚

𝑛

= ∑ ∑ |𝛥(𝑠𝑖−1 , 𝑡𝑗−1 ) − 𝛥(𝑠𝑖 , 𝑡𝑗 ) − 𝛥(𝑠𝑖−1 , 𝑡𝑗 ) + 𝛥(𝑠𝑖 , 𝑡𝑗−1 )| 𝑖=1 𝑗=1 𝑚

𝑛

∞ ∞

≤ 4 ∑ ∑ |𝛥(𝑠𝑖 , 𝑡𝑗 ))| ≤ 4 ∑ ∑ |𝛥(𝑟𝑘 , 𝑟𝑙 ))| 𝑖=0 𝑗=0

𝑘=1 𝑙=1

∞ ∞

|ℎ (𝑓(𝑟𝑘 , 𝑟𝑙 )) − ℎ0 (𝑔(𝑟𝑘 , 𝑟𝑙 ))| =4∑∑ 0 2𝑘+𝑙 𝑘=1 𝑙=1 ∞ ∞

≤ 4𝐿 ∑ ∑ 𝑘=1 𝑙=1

|(𝑓 − 𝑔)(𝑟𝑘 , 𝑟𝑙 )| ≤ 4𝐿‖𝑓 − 𝑔‖𝐵𝑉 , 2𝑘+𝑙

which shows that V2 (𝑆ℎ 𝑓 − 𝑆ℎ 𝑔; [0, 1] × [0, 1]) ≤ 4𝐿‖𝑓 − 𝑔‖𝐵𝑉 . The variation Var((𝑆ℎ 𝑓 − 𝑆ℎ 𝑔)(⋅, 0), 𝑃; [0, 1]) occurring in (1.76) may be estimated in the form 𝑚

Var((𝑆ℎ 𝑓 − 𝑆ℎ 𝑔)(⋅, 0), 𝑃; [0, 1]) = ∑ |𝛥(𝑠𝑖 , 0) − 𝛥(𝑠𝑖−1 , 0)| 𝑖=1 ∞

|ℎ0 (𝑓(𝑟𝑘 , 0)) − ℎ0 (𝑔(𝑟𝑘 , 0))| ≤ 4𝐿‖𝑓 − 𝑔‖𝐵𝑉 , 2𝑘 𝑘=1

≤4∑

6.5 Comments on Chapter 6

| 419

and similarly for the variation Var((𝑆ℎ 𝑓 − 𝑆ℎ 𝑔)(0, ⋅), 𝑄; [0, 1]) occurring in (1.77). Finally, noting that |(𝑆ℎ 𝑓 − 𝑆ℎ 𝑔)(0, 0)| = |𝛥(0, 0)| ≤ |ℎ0 (𝑓(0, 0)) − ℎ0 (𝑔(0, 0))| ≤ 𝐿|𝑓(0, 0) − 𝑔(0, 0)| , and combining this with the previous estimates, we conclude that ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖BV = |𝛥(0, 0)| + Var(𝛥; [0, 1] × [0, 1]) = |𝛥(0, 0)| + Var(𝛥(⋅, 0); [0, 1]) + Var(𝛥(0, ⋅); [0, 1]) + V2 (𝛥; [0, 1] × [0, 1]) ≤ 𝐿|𝑓(0, 0) − 𝑔(0, 0)| + 2𝐿 Var(𝑓(⋅, 0) − 𝑔(⋅, 0); [0, 1]) + 2𝐿 Var(𝑓(0, ⋅) − 𝑔(0, ⋅); [0, 1]) + 4𝐿 V2 (𝑆ℎ 𝑓 − 𝑆ℎ 𝑔; [0, 1] × [0, 1]) ≤ 𝐿|(𝑓 − 𝑔)(0, 0)| + 4𝐿 Var(𝑓 − 𝑔; [0, 1] × [0, 1]) ≤ 4𝐿‖𝑓 − 𝑔‖BV , which proves (6.30) with 𝐾 = 4𝐿, although ℎ is not of the form (6.69).

♥

6.5 Comments on Chapter 6 Some of the material treated in Section 6.1 may be found in the monograph [20]. Exam­ ple 6.3 is taken from [48], see also [46, 47], Theorem 6.4 and Proposition 6.5 from [52], for proofs, see [53]. Theorem 6.7 coincides with [20, Theorem 8.1], where this result is apparently stated for the first time. The problem to describe sufficient conditions on ℎ (which are possibly close to be­ ing necessary) under which the corresponding operator 𝑆ℎ maps 𝐵𝑉 into itself has a long history. Lyamin [189] has given (without proof) a sufficient condition, but Buga­ jewska expressed in her paper [66] some doubt about the correctness of this condi­ tion. Her objection was fully justified by Maćkowiak’s surprising counterexample [190] which is our Example 6.8. Afterwards, Bugajewska gave a correct sufficient condition, together with an illuminating discussion, in [66]; this discussion is rephrased in our Theorems 6.11 and 6.13 and in Example 6.12. The necessary condition contained in Theorem 6.10 may be found in an implicit form in the recent beautiful book [107]; we have “extracted” the main idea in the proof of Theorem 6.10. Of course, there are still many open questions. Among them, we mention the fol­ lowing problem related to 𝐵𝑉-functions. Problem 6.1. Let ℎ : [𝑎, 𝑏] × ℝ → ℝ be continuous, and suppose that the corresponding operator (6.1) maps the space 𝐵𝑉([𝑎, 𝑏]) into itself and is both bounded and continuous. Does this imply that ℎ(𝑥, ⋅) ∈ 𝐿𝑖𝑝loc (ℝ) for every⁹ 𝑥 ∈ [𝑎, 𝑏]? 9 Observe that the function ℎ in Example 6.9 is discontinuous at each point (0, 𝑢) for 𝑢 ≠ 0.

420 | 6 Nonlinear superposition operators Problem 6.2. For 𝑋 ∈ {𝐴𝐶, 𝑅𝐵𝑉𝑝 }, find ℎ : [𝑎, 𝑏] × ℝ → ℝ such that 𝑆ℎ (𝑋) ⊆ 𝑋, but ℎ(𝑥, ⋅) ∈ ̸ 𝐿𝑖𝑝loc (ℝ) for some 𝑥 ∈ [𝑎, 𝑏]. We remark that for 𝑋 ∈ {𝐵𝑉, 𝑊𝐵𝑉𝑝 }, we have the counterexample (6.14), while for 𝑋 = 𝐿𝑖𝑝 and 𝑋 = 𝐿𝑖𝑝𝛼 , we have the counterexample (6.4). Problem 6.3. Find ℎ : [𝑎, 𝑏] × ℝ → ℝ such that 𝑆ℎ (Λ𝐵𝑉) ⊆ Λ𝐵𝑉, but ℎ(𝑥, ⋅) ∈ ̸ 𝐿𝑖𝑝loc (ℝ) for some 𝑥 ∈ [𝑎, 𝑏]. What about boundedness and/or continuity of 𝑆ℎ ? Concerning spaces of differentiable function, the following example which is paral­ lel to Example 6.9 shows that the inclusion 𝑆ℎ (𝐵𝑉1 ) ⊆ 𝐵𝑉1 does not imply that the function ℎ(𝑥, ⋅) belongs to 𝐿𝑖𝑝1loc (ℝ) for every 𝑥 ∈ [𝑎, 𝑏]. Example 6.36. Define ℎ0 : ℝ → ℝ by 0 { { {2 ℎ0 (𝑢) := { 3 𝑢√𝑢 { { 1 {𝑢 − 3

if 𝑢 ≤ 0 , if 0 < 𝑢 < 1 , if 𝑢 ≥ 1 ,

and ℎ : [0, 1] × ℝ → ℝ by (6.14). Then the corresponding operator (6.1) maps the space 𝐵𝑉1 ([0, 1]) into itself and is bounded. To see this, fix 𝑓 ∈ 𝐵𝑉1 ([0, 1]) and observe that the function 𝑔 = 𝑆ℎ 𝑓 satisfies 𝑔󸀠 (0) = ℎ󸀠0 (𝑓(0))𝑓󸀠 (0),

𝑔󸀠 (𝑥) = 𝑔(𝑥) ≡ 0

(𝑥 > 0) .

Consequently, Var(𝑔󸀠 , 𝑃; [0, 1]) = |ℎ󸀠0 (𝑓(0))𝑓󸀠 (0)| ≤ |𝑓󸀠 (0)| ≤ ‖𝑓‖𝐵𝑉1 for every partition 𝑃 = {𝑡0 , 𝑡1 , . . . , 𝑡𝑚 } ∈ P([0, 1]). On the other hand, the function ℎ(0, ⋅) = ℎ0 does not belong to 𝐿𝑖𝑝1loc (ℝ) since ℎ󸀠0 (𝑢) = √𝑢 is not locally Lipschitz at zero. ♥ For other spaces of differentiable function,¹⁰ the corresponding problem seems to be open: Problem 6.4. For 𝑋 ∈ {𝐿𝑖𝑝, 𝐿𝑖𝑝𝛼 , 𝐴𝐶, 𝑅𝐵𝑉𝑝 }, find ℎ : [𝑎, 𝑏] × ℝ → ℝ such that 𝑆ℎ (𝑋1 ) ⊆ 𝑋1 , but ℎ(𝑥, ⋅) ∈ ̸ 𝐿𝑖𝑝1loc (ℝ) for some 𝑥 ∈ [𝑎, 𝑏]. Problem 6.5. For 𝑋 ∈ {𝐿𝑖𝑝, 𝐿𝑖𝑝𝛼 , 𝐴𝐶, 𝐵𝑉, 𝑊𝐵𝑉𝑝 , 𝑅𝐵𝑉𝑝 }, find conditions on ℎ : [𝑎, 𝑏] × ℝ → ℝ, possibly both necessary and sufficient, under which 𝑆ℎ is bounded and/or con­ tinuous in 𝑋1 . Concerning the Wiener–Young space 𝑊𝐵𝑉𝜙 , we also repeat the following open prob­ lem which we stated after the proof of Theorem 6.13:

10 In case 𝑋1 = 𝐿𝑖𝑝1 and 𝑋1 = 𝐿𝑖𝑝1𝛼 , one could try to adapt the proof of Theorem 6.4.

6.5 Comments on Chapter 6

|

421

Problem 6.6. Is Theorem 6.13 true without the requirement 𝜙 ∈ 𝛿2 ? In the recent paper [35], it is shown that the operator 𝑆ℎ maps the space 𝑅([𝑎, 𝑏]) of regular functions into itself if ℎ(𝑥, ⋅) is continuous, uniformly with respect to 𝑥 ∈ [𝑎, 𝑏], and ℎ(⋅, 𝑢) is regular for all 𝑢 ∈ ℝ. Moreover, in this case, 𝑆ℎ is bounded and continuous in the norm (0.39). The paper [35] also contains a counterexample which shows that these conditions on ℎ are not necessary for 𝑆ℎ (𝑅) ⊆ 𝑅. We remark that [310] proves an analogous result, but assuming a priori that ℎ(𝑥, ⋅) ∈ 𝐿𝑖𝑝loc (ℝ). The Matkowski property and weak Matkowski property is discussed in detail in the book [226]. What we call a uniform (weak) Matkowski property, however, has been analyzed only quite recently in a series of papers. At the risk of being redundant, let us repeat the degeneracy list which we stated at the beginning of Section 5.4, but for the superposition operator (6.1), and in extended form, including the uniform (weak) Matkowski property. It was shown – in [193] that the space 𝐿𝑖𝑝𝛼 ([𝑎, 𝑏]) of Hölder continuous functions of order 𝛼 < 1 with norm (0.71) has the Matkowski property, and in [201], that it even has the uniform Matkowski property:¹¹ – in [206] that the space 𝐶𝑛 ([𝑎, 𝑏]) of 𝑛-times continuously differentiable functions with norm (0.63) has the Matkowski property, and in [200], that it even has the uniform Matkowski property; – in [194] that the space 𝐴𝐶([𝑎, 𝑏]) of absolutely continuous functions with norm (3.42) has the Matkowski property, and in [124], that it even has the uniform Matkowski property; – in [162] that the space 𝐿𝑖𝑝𝑛 ([𝑎, 𝑏]) of functions with Lipschitz continuous 𝑛-th derivative with norm (0.78) has the Matkowski property; – in [187] that the space 𝐿𝑖𝑝𝑛𝛼 ([𝑎, 𝑏]) of functions with Hölder continuous 𝑛-th deriva­ tive has the Matkowski property; – in [289] that the space 𝐴𝐶𝑛 ([𝑎, 𝑏]) of functions with absolutely continuous 𝑛-th derivative has the Matkowski property; – in [205] and [224] that the space 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) = 𝑊𝑝1 ([𝑎, 𝑏]) of functions of bounded 𝑝-variation in Riesz’s sense for 1 < 𝑝 < ∞ with norm (2.90) has the Matkowski property; – in [205] that the higher order Sobolev space 𝑊𝑝𝑛 ([𝑎, 𝑏]) has the Matkowski prop­ erty; – in [204] that the space 𝑅𝐵𝑉2,𝑝 ([𝑎, 𝑏]) of functions of bounded (2, 𝑝)-variation in Riesz’s sense with norm (3.77) has the Matkowski property;

11 A generalization to abstract (i.e. Banach space valued) Lipschitz continuous functions may be found in [203].

422 | 6 Nonlinear superposition operators –

– –

– – – –

in [217] that the space 𝑅𝐵𝑉𝜙 ([𝑎, 𝑏]) of functions of bounded Riesz–Medvedev vari­ ation with norm (2.99) has the Matkowski property, and in [1], that it even has the uniform Matkowski property;¹² in [32] that the space 𝜅𝐵𝑉([𝑎, 𝑏]) of functions of bounded Korenblum variation with norm (2.136) has the Matkowski property; in [207] that the space 𝐵𝑉([𝑎, 𝑏]) of functions of bounded variation with norm (1.16) has the weak Matkowski property, and in [202], that it even has the uniform weak Matkowski property; in [224] that the space 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) of functions of bounded Wiener 𝑝-variation with norm (1.65) has the uniform weak Matkowski property; in [135] that the space 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) of functions of bounded Wiener–Young varia­ tion with norm (2.11) has the uniform weak Matkowski property;¹³ in [98] that the space Λ𝐵𝑉([𝑎, 𝑏]) of functions of bounded Waterman variation with norm (2.30) has the weak Matkowski property; in [112] that the space 𝛷𝐵𝑉([𝑎, 𝑏]) of functions of bounded Schramm variation with norm (2.75) has the uniform weak Matkowski property.

Recent results connected with the Matkowski or weak Matkowski property may be found in [1, 8, 22, 29–31, 57, 110, 119, 123, 124, 137, 151, 152, 197–199, 208, 233]. In con­ nection with degeneracy phenomena, we state another open question which seems to be difficult and interesting: Problem 6.7. Does there exist a pair (𝑋, 𝑌) of Banach spaces 𝑋 and 𝑌 having the Matkowski property, but not the uniform Matkowski property? More specifically, can we construct two function spaces 𝑋 and 𝑌 over [0, 1], say, such that Lipschitz con­ tinuous superposition operators 𝑆ℎ : 𝑋 → 𝑌 are generated only by affine functions ℎ : [0, 1] × ℝ → ℝ, but there is some nonaffine function ℎ̂ : [0, 1] × ℝ → ℝ such that the corresponding operator 𝑆ℎ̂ : 𝑋 → 𝑌 is uniformly bounded? In [203], the following alternative definition of uniform boundedness is given. A (usu­ ally, nonlinear) operator 𝐴 : 𝑋 → 𝑌 is called uniformly bounded if there exists a function 𝛾 : [0, ∞) → [0, ∞) such that 𝛾(0) = 0, and for every nonempty set 𝑀 ⊂ 𝑋 from 𝑑𝑖𝑎𝑚(𝑀) ≤ 𝑡, it follows that 𝑑𝑖𝑎𝑚(𝐴(𝑀)) ≤ 𝛾(𝑡). It is not hard to see, however, that this is equivalent to uniform boundedness in the sense of our Definition 6.22. To see this, suppose that 𝐴 : 𝑋 → 𝑌 satisfies (6.49), and let 𝑑𝑖𝑎𝑚(𝑀) ≤ 𝑡 for some bounded set 𝑀 ⊂ 𝑋. Then ‖𝑓 − 𝑔‖ ≤ 𝑡 for all 𝑓, 𝑔 ∈ 𝑀, and so ‖𝐴𝑓 − 𝐴𝑔‖ ≤ 𝛾(‖𝑓 − 𝑔‖) ≤ 𝛾(𝑡)

12 For the same result in the space 𝑅𝐵𝑉𝜙,𝑤 ([𝑎, 𝑏]) with weighted Riesz variation (2.160), see [27], and for multivalued functions see [28, 29]. 13 An analogous result for functions of two variables may be found in [136], for multivalued functions in [36].

6.6 Exercises to Chapter 6 | 423

by the monotonicity of 𝛾. Since 𝑓, 𝑔 ∈ 𝑀 are arbitrary, this precisely means that 𝑑𝑖𝑎𝑚(𝐴(𝑀)) ≤ 𝛾(𝑡). Conversely, suppose that 𝑑𝑖𝑎𝑚(𝑀) ≤ 𝑡 implies 𝑑𝑖𝑎𝑚(𝐴(𝑀)) ≤ 𝛾(𝑡), where 𝑀 ⊂ 𝑋 is bounded. Applying this, in particular, to the set 𝑀 := {𝑓, 𝑔}, we see that (6.49) is true, and so 𝐴 is uniformly bounded in the sense of Definition 6.22. The advantage of the boundedness condition in the above definition consists of its great generality: in fact, this definition carries over to nonlinear operators in arbitrary metric spaces. Example 6.24 is explained by the fact that the superposition operator 𝑆ℎ is contin­ uous between 𝐿 1 and 𝐿 ∞ only if 𝑆ℎ is constant, i.e. (6.32) holds with 𝛽(𝑥) ≡ 0. A proof of this surprising fact can be found in [324] or [20, Theorem 3.17], see also Exercise 6.2.

6.6 Exercises to Chapter 6 We state some exercises on the topics covered in this chapter; exercises marked with an asterisk * are more difficult. Exercise 6.1. Show that the operator 𝑆ℎ maps 𝐿 𝑝 ([𝑎, 𝑏]) (1 ≤ 𝑝 < ∞) into 𝐿 ∞ ([𝑎, 𝑏]) if and only if |ℎ(𝑥, 𝑢)| ≤ 𝛼(𝑥) for some 𝛼 ∈ 𝐿 ∞ ([𝑎, 𝑏]); moreover, in this case, 𝑆ℎ is automatically bounded. Exercise 6.2. Show that the operator 𝑆ℎ is continuous between 𝐿 𝑝 ([𝑎, 𝑏]) (1 ≤ 𝑝 < ∞) and 𝐿 ∞ ([𝑎, 𝑏]) if and only if ℎ(𝑥, 𝑢) = 𝛼(𝑥) for some 𝛼 ∈ 𝐿 ∞ ([𝑎, 𝑏]), i.e. 𝑆ℎ is constant. Exercise 6.3. Show that the operator 𝑆ℎ maps 𝐿 ∞ ([𝑎, 𝑏]) into 𝐿 𝑞 ([𝑎, 𝑏]) (1 ≤ 𝑞 < ∞) if and only if for each 𝑟 > 0 there exists a function 𝛼𝑟 ∈ 𝐿 𝑞 ([𝑎, 𝑏]) such that |ℎ(𝑥, 𝑢)| ≤ 𝛼𝑟 (𝑥)

(𝑎 ≤ 𝑥 ≤ 𝑏, |𝑢| ≤ 𝑟) ;

moreover, in this case, 𝑆ℎ is automatically bounded and continuous. Exercise 6.4. Show that the operator 𝑆ℎ maps 𝐿 ∞ ([𝑎, 𝑏]) into 𝐿 ∞ ([𝑎, 𝑏]) if and only if for each 𝑟 > 0, there exists a function 𝛼𝑟 ∈ 𝐿 ∞ ([𝑎, 𝑏]) such that the condition from Exercise 6.3 holds; moreover, in this case, 𝑆ℎ is automatically bounded. Exercise 6.5. Show that the operator 𝑆ℎ is continuous in 𝐿 ∞ ([𝑎, 𝑏]) if and only if for each 𝑟 > 0, there exists a continuous function 𝛽𝑟 : [0, ∞) → [0, ∞) such that 𝛽𝑟 (0) = 0 and |ℎ(𝑥, 𝑢) − ℎ(𝑥, 𝑣)| ≤ 𝛽𝑟 (|𝑢 − 𝑣|) (𝑎 ≤ 𝑥, ≤ 𝑏, |𝑢|, |𝑣| ≤ 𝑟) . Exercise 6.6. Suppose that ℎ : [𝑎, 𝑏]×ℝ → ℝ has the property that the corresponding superposition operator (6.1) maps a function space 𝑋 into itself and is bounded in the norm of 𝑋. In analogy to Exercise 5.11, for 𝑟 > 0, we put 𝜇𝑟 (ℎ, 𝑋) := sup {‖𝑆ℎ 𝑓‖𝑋 : ‖𝑓‖𝑋 ≤ 𝑟}

424 | 6 Nonlinear superposition operators and call this characteristic the growth function of ℎ in 𝑋. Calculate the functions 𝜇𝑟 (ℎ, 𝐶) and 𝜇𝑟 (ℎ, 𝐶1 ) under the hypotheses of Theorem 6.1 and Theorem 6.7, respec­ tively. Exercise 6.7. Using the notation of Theorem 6.4, show that the growth function 𝜇𝑟 (ℎ, 𝐿𝑖𝑝𝛼 ) from Exercise 6.6 satisfies the bilateral estimate 𝑘(𝑟) ≤ 𝜇𝑟 (ℎ, 𝐿𝑖𝑝𝛼 ) ≤ 𝑘(𝑟) , +1

2𝛼+1

where 𝑘(𝑟) is the local Hölder–Lipschitz constant occurring in (6.5). Exercise 6.8. In analogy to (6.65), define the right-right regularization 𝑓##, the right-left regularization 𝑓#♭ , and the left-right regularization 𝑓♭# of 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ, and show that Theorem 6.34 also holds for the regularizations ℎ##(⋅, ⋅, 𝑢), ℎ#♭ (⋅, ⋅, 𝑢), and ℎ♭#(⋅, ⋅, 𝑢). Exercise 6.9. Let [𝑎1 , 𝑏1 ], [𝑎2 , 𝑏2 ], . . . , [𝑎𝑛 , 𝑏𝑛 ] be real intervals, and denote by 𝐼𝑎𝑏 := [𝑎1 , 𝑏1 ] × [𝑎2 , 𝑏2 ] × . . . × [𝑎𝑛 , 𝑏𝑛 ] ⊂ ℝ𝑛 their Cartesian product. In analogy to (1.82), introduce some kind of variation Var(𝑓, 𝑃1 × 𝑃2 × . . . × 𝑃𝑛 ; 𝐼𝑎𝑏 ) for a function 𝑓 : 𝐼𝑎𝑏 → ℝ with respect to partitions 𝑃𝑘 ∈ P([𝑎𝑘 , 𝑏𝑘 ]) (𝑘 = 1, 2, . . . , 𝑛). Given a function of 𝑛 + 1 variables ℎ : 𝐼𝑎𝑏 × ℝ → ℝ, by imitating The­ orem 6.11 formulate and prove a sufficient condition under which the superposition operator 𝑆ℎ 𝑓(𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) := ℎ(𝑥1 , 𝑥2 , . . . , 𝑥𝑛 , 𝑓(𝑥1 , 𝑥2 , . . . , 𝑥𝑛 )) generated by this function maps the corresponding space 𝐵𝑉(𝐼𝑎𝑏 ) into itself and is bounded.

7 Some applications This section is concerned with a few applications to problems which lead in a natu­ ral way to spaces of functions of bounded (generalized) variation. Historically, con­ vergence criteria for Fourier series have always been one of the main motivations for introducing new concepts of variation. Below, we show that functions from the Water­ man space Λ𝐵𝑉 are in a certain sense “optimal” for being represented by their Fourier series because they produce sharp convergence results. In the second part of this chap­ ter, we show how our results on composition and superposition operators from the preceding two chapters may be used to prove existence (and even uniqueness) of so­ lutions to nonlinear integral equations with regular or weakly singular kernels.

7.1 Convergence criteria for Fourier series Almost two centuries ago, Dirichlet [104] proved that every piecewise monotone real function on an interval has a pointwise convergent Fourier series. This result is usually referred to as the Dirichlet criterion and may be considered as a first contribution to a rigorous (partial) proof of Fourier’s famous (wrong) conjecture [115], raised in 1807, on the possibility to expand arbitrary real functions into a trigonometric series. Ac­ cording to Szőkefalvi–Nagy [301], the history of Fourier series started with a fruitful controversial discussion in the middle of the nineteenth century between D’Alembert, Euler, and D. Bernoulli regarding the problem of the vibrating string. An important progress was achieved in 1881 by Jordan [153] who not only intro­ duced functions of bounded variation, but also proved that such functions may be represented as differences of increasing functions, in this way extending the validity of Dirichlet’s result to 𝐵𝑉-functions. Subsequently, convergence criteria for Fourier se­ ries have been one of the main motivations for introducing and studying new concepts of variation. The purpose of this and the following section is to present some sample results in this spirit. In this section, we always consider functions 𝑓 defined on [𝑎, 𝑏] = [0, 2𝜋] and satisfying 𝑓(0) = 𝑓(2𝜋); thus, they may be extended periodically to the whole real line. Let 𝑓 : [0, 2𝜋] → ℝ be continuous of period 2𝜋. The (real) Fourier series of 𝑓 is given by ∞ 𝛼 𝑆[𝑓] = 𝑆[𝑓](𝑥) = 0 + ∑ [𝛼𝑛 cos 𝑛𝑥 + 𝛽𝑛 sin 𝑛𝑥] , (7.1) 2 𝑛=1 where

2𝜋

𝛼𝑛 = 𝛼𝑛 (𝑓) =

1 ∫ 𝑓(𝑡) cos 𝑛𝑡 𝑑𝑡 , 𝜋 0

(𝑛 = 0, 1, 2, . . .)

(7.2)

426 | 7 Some applications and

2𝜋

1 𝛽𝑛 = 𝛽𝑛 (𝑓) = ∫ 𝑓(𝑡) sin 𝑛𝑡 𝑑𝑡 𝜋

(𝑛 = 1, 2, 3, . . .)

(7.3)

0

are the (real) Fourier coefficients of 𝑓. A crucial problem in the theory and applications of Fourier series consists of finding conditions under which the series (7.1) converges pointwise, uniformly, or in some other sense.¹ It is well known that the partial sums 𝑠𝑛 (𝑥) = 𝑠𝑛 (𝑥; 𝑓) =

𝑛 𝛼0 + ∑ [𝛼𝑘 cos 𝑘𝑥 + 𝛽𝑘 sin 𝑘𝑥] 2 𝑘=1

(7.4)

of (7.1) may be written as convolution 2𝜋

𝑠𝑛 (𝑥) = (𝐷𝑛 ∗ 𝑓)(𝑥) = ∫ 𝐷𝑛 (𝑥 − 𝑡)𝑓(𝑡) 𝑑𝑡

(7.5)

0

of 𝑓 and the so-called Dirichlet kernel 𝑛

𝐷𝑛(𝑥) = 1 + 2 ∑ cos 𝑘𝑥 = 𝑘=1

sin((𝑛 + 1/2)𝑥) sin(𝑥/2)

(𝑛 = 0, 1, 2, . . .) .

(7.6)

If instead of (7.4) we consider the average of the partial sums, i.e. 𝑠0 (𝑥) + 𝑠1 (𝑥) + . . . + 𝑠𝑛 (𝑥) , (7.7) 𝑛+1 we may write 𝜎𝑛 as convolution 𝜎𝑛 = 𝐾𝑛 ∗ 𝑓 of 𝑓 and the so-called Féjèr kernels 𝐾𝑛 . While the Dirichlet kernels 𝐷𝑛 are oscillatory, the Féjèr kernels are positive. More pre­ cisely, one may show that 𝜎𝑛 (𝑥) = 𝜎𝑛 (𝑥; 𝑓) =

2𝜋

2𝜋

∫ 𝐹𝑛 (𝑡) 𝑑𝑡 = ∫ |𝐹𝑛 (𝑡)| 𝑑𝑡 ≡ 1 , 0

but

0 2𝜋

2𝜋

∫ 𝐷𝑛(𝑡) 𝑑𝑡 ≡ 1, 0

(7.8)

lim ∫ |𝐷𝑛(𝑡)| 𝑑𝑡 = ∞ .

𝑛→∞

(7.9)

0

This is one of the reasons why the partial sums 𝑠𝑛 present more difficulties² than their averages 𝜎𝑛 , which behave quite well. For instance, the sequence (𝐾𝑛 (𝑡))𝑛 con­ verges uniformly to zero on ℝ \ (−𝛿, 𝛿) for every 𝛿 > 0, which implies that (𝜎𝑛 (𝑥; 𝑓))𝑛 converges uniformly to 𝑓, as 𝑛 → ∞, for each continuous function 𝑓.

1 The estimates (4.49) and (4.51) in Proposition 4.23 show that the Fourier coefficients of a 𝐵𝑉-function converge to zero; of course, this is only necessary for the convergence of (7.1). 2 In fact, one may consider the map ℓ𝑛 (𝑓) := 𝑠𝑛 (0; 𝑓) for each 𝑛 as a bounded linear functional on the space of 2𝜋-periodic continuous functions. From the functional-analytic Banach–Steinhaus theorem and the unboundedness of the sequence (𝐷𝑛 )𝑛 in the 𝐿 1 -norm, see (7.9), we may then conclude that the sequence (ℓ𝑛 (𝑓))𝑛 is also unbounded for some continuous 𝑓, and so the Fourier series of 𝑓 is accordingly divergent at zero.

7.1 Convergence criteria for Fourier series |

427

The existence of continuous functions 𝑓 whose Fourier series diverge at some point emphasizes the need of imposing additional conditions which guarantee the convergence of 𝑆[𝑓]. Such conditions are sometimes called convergence criteria or con­ vergence tests. We recall some of them which will be needed in the sequel. To state one criterion of particular historical interest, we use the conventional notation 𝜑(𝑡) = 𝜑𝑥 (𝑡) :=

𝑓(𝑥 + 𝑡) + 𝑓(𝑥 − 𝑡) − 2𝑓(𝑥) 2

(0 ≤ 𝑡 ≤ 2𝜋)

(7.10)

throughout this section, where 𝑥 ∈ [0, 2𝜋] is fixed. – The Lebesgue test. Suppose that ℎ

∫ |𝜑𝑥 (𝑡)| 𝑑𝑡 = 𝑜(ℎ)

(ℎ → 0)

(7.11)

0

and

2𝜋

∫ 𝜂𝑛

|𝜑𝑥 (𝑡) − 𝜑𝑥 (𝑡 + 𝜂𝑛 )| 𝑑𝑡 = 𝑜(1) 𝑡

(𝑛 → ∞) ,

(7.12)

where 𝜂𝑛 := 2𝜋/𝑛. Then the Fourier series (7.1) converges to 𝑓(𝑥). Moreover, the con­ vergence is uniform over any closed interval of continuity of 𝑓, where condition (7.12) is satisfied uniformly. The proof of the Lebesgue test can be found in any book on trigonometric series or harmonic analysis, e.g. [329]. A particularly interesting sufficient condition given in [240] and refined in [49] asserts that one may reach convergence of Fourier series by a suitable homeomorphic change of variables: – The Pál–Bohr test. Let 𝑔 : [0, 2𝜋] → ℝ be continuous and 2𝜋-periodic. Then there exists a homeomorphism 𝜏 : [0, 2𝜋] → [0, 2𝜋] such that the Fourier series (7.1) of 𝑓 = 𝑔 ∘ 𝜏 converges uniformly. The following criteria refer to functions of bounded variation and related classes. They are are due to Jordan [153], Salem [285], Lipschitz and Dini [329], Goffman and Water­ man [132], Garcia and Sawyer [116], and Sahney and Waterman [284], respectively. In particular, the Garcia–Sawyer test makes use of the Banach indicatrix 𝐼𝑓 of 𝑓 intro­ duced in (0.106). Recall that 𝑓 and 𝐼𝑓 are related by the equality ∞

Var(𝑓; [0, 2𝜋]) = ∫ 𝐼𝑓 (𝑦) 𝑑𝑦 , −∞

and so 𝑓 ∈ 𝐵𝑉([0, 2𝜋]) if and only if 𝐼𝑓 ∈ 𝐿 1 (ℝ), see Proposition 1.27. – The Jordan test. Let 𝑓 : [0, 2𝜋] → ℝ be continuous, of bounded variation, and 2𝜋-periodic. Then the Fourier series (7.1) converges uniformly on [0, 2𝜋] to 𝑓(𝑥).

428 | 7 Some applications –

The Salem test. Let 𝑓 : [0, 2𝜋] → ℝ be continuous and 2𝜋-periodic. For 𝑛 ∈ ℕ, put 𝜏1 :=

2𝜋 4𝜋 2(𝑛 − 1)𝜋 , 𝜏 := , . . . , 𝜏𝑛−1 := , 𝜏𝑛 := 2𝜋 𝑛 2 𝑛 𝑛

and

(𝑛+1)/2

𝑇𝑛 (𝑥) := ∑ 𝑘=1

–

–

–

(7.13)

𝑓(𝑥 + 𝜏2𝑘−2 ) − 𝑓(𝑥 + 𝜏2𝑘−1 ) . 𝑘

If the sequence (𝑇𝑛 )𝑛 converges uniformly to zero as 𝑛 → ∞, the Fourier series (7.1) of 𝑓 converges uniformly. The Lipschitz–Dini test. Let 𝑓 : [0, 2𝜋] → ℝ be continuous and 2𝜋-periodic. As­ sume that 1 ) (𝛿 → 0) , (7.14) 𝜔∞ (𝑓; 𝛿) = 𝑜 ( log 𝛿 where 𝜔∞ (𝑓; ⋅) denotes the modulus of continuity (0.97) of 𝑓. Then the Fourier series (7.1) of 𝑓 converges uniformly. The Goffman–Waterman test. Let 𝜙 : [0, ∞) → [0, ∞) be some Young func­ tion, and let 𝜙∗ denote its conjugate Young function (0.23). Let 𝑓 ∈ 𝐶([0, 2𝜋]) ∩ 𝑊𝐵𝑉𝜙 ([0, 2𝜋]) be 2𝜋-periodic, where 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) is the Wiener–Young space in­ troduced in Definition 2.2. If ∞ 1 (7.15) ∑ 𝜙∗ ( ) < ∞ , 𝑛 𝑛=1 the Fourier series (7.1) of 𝑓 converges uniformly. The Garsia–Sawyer test. Let 𝑓 : [0, 2𝜋] → ℝ be continuous and 2𝜋-periodic. If ∞

(7.16)

∫ log 𝐼𝑓 (𝑦) 𝑑𝑦 < ∞ , −∞

–

the Fourier series (7.1) of 𝑓 converges uniformly. The Sahney–Waterman test. Suppose that the function 𝜑𝑥 defined by (7.10) satis­ fies (7.11) and 2𝜋

∫ 𝜂𝑛

|𝜑𝑥 (𝑡) − 𝜑𝑥 (𝑡 + 𝜂𝑛 )| 𝑑𝑡 = 𝑜(𝜂𝑛−𝛽 ) 𝑡𝛽+1

(𝑛 → ∞)

(7.17)

for some 𝛽 ∈ (−1, 0), where 𝜂𝑛 :=

2𝜋 𝑛+

1 (𝛽 2

+ 1)

.

(7.18)

Then the Fourier series (7.1) satisfies 𝑆[𝑓](𝑥) = 𝑜(𝑛−𝛽 ) and converges to 𝑓(𝑥).

(7.19)

7.2 Fourier series and Waterman spaces | 429

We point out that these criteria are not independent of each other, and some particular cases of them have been known earlier. For example, letting 𝜙(𝑡) = 𝑡𝑝 (1 < 𝑝 < ∞) in 󸀠 the Goffman–Waterman test, we obtain 𝜙∗ (𝑡) = 𝑡𝑝 = 𝑡𝑝/(𝑝−1) (up to constants), and so (7.15) reads ∞ 1 ∑ 𝑝/(𝑝−1) < ∞ . (7.20) 𝑛=1 𝑛 This condition was already known to Young [323]. Interestingly, condition (7.15) also implies that 𝑊𝐵𝑉𝜙 ([𝑎, 𝑏]) ⊆ 𝐻𝐵𝑉([𝑎, 𝑏]), the space of functions of bounded har­ monic variation, see Definition 2.29. In fact, this follows from Proposition 2.34 for the particular choice 𝜆 𝑛 = 1/𝑛. Also, the Salem criterion extends the Lipschitz–Dini test. Indeed, if 𝜔∞ (𝑓; ⋅) is the modulus of continuity (0.97) of 𝑓, it is not hard to see that |𝑇𝑛(𝑥)| = 𝑂(𝜔∞ (𝑓; 2𝜋/𝑛) log 𝑛)

(𝑛 → ∞) .

(7.21)

Combining this with condition (7.14) and applying the Salem criterion, we get the uniform convergence of the Fourier series (7.1). As a second application of the Salem criterion, we obtain the Jordan condition 𝑓 ∈ 𝐵𝑉([0, 2𝜋]) ∩ 𝐶([0, 2𝜋]). For such functions 𝑓, we have |𝑇𝑛 (𝑥)| ≤ 𝜔∞ (𝑓; 2𝜋/𝑛) (1 +

Var(𝑓; [0, 2𝜋]) 1 1 + ... + ) + , 2 𝑚 𝑚+1

where 𝑚 is any integer smaller than (𝑛+1)/2. If we choose 𝑚, as we may, so that 𝑚 goes to infinity, but 𝜔∞ (𝑓; 2𝜋/𝑛) log 𝑚 goes to zero, as 𝑛 → ∞, we get again the uniform con­ vergence of the Fourier series (7.1). Also, we get the Jordan test from the Garcia–Sawyer test since the integral (7.16) is finite for 𝑓 ∈ 𝐵𝑉([0, 2𝜋]) ∩ 𝐶([0, 2𝜋]).

7.2 Fourier series and Waterman spaces In this section, we take a closer look at the Fourier series of functions in the Waterman spaces Λ𝐵𝑉 which we introduced and discussed in detail in Section 2.2. The first result in this direction [314] is concerned with absolute convergence of the Fourier series (7.1). Theorem 7.1. Let Λ = (𝜆 𝑛 )𝑛 be a Waterman sequence, and let 𝑓 ∈ 𝐶([0, 2𝜋]) ∩ Λ𝐵𝑉([0, 2𝜋]) be 2𝜋-periodic. Suppose that ∞

∑ 𝑛=1

√𝜔∞ (𝑓; 2𝜋/𝑛) 𝑛√𝜆 𝑛

< ∞,

(7.22)

where 𝜔∞ (𝑓; ⋅) denotes the modulus of continuity (0.97) of 𝑓 and the convergence in (7.22) is monotone. Then the Fourier series (7.1) of 𝑓 converges absolutely. Proof. Fix 𝑓 ∈ 𝐶([0, 2𝜋]) ∩ Λ𝐵𝑉([0, 2𝜋]), define numbers 𝜏𝑘 as in (7.13), and let 𝑎𝑘 := 𝑥 + 𝜏𝑘−1 and 𝑏𝑘 := 𝑥 + 𝜏𝑘 . For the collection 𝑆 := {[𝑎1 , 𝑏1 ], . . . , [𝑎𝑛 , 𝑏𝑛 ]} ∈ 𝛴([0, 2𝜋]), we

430 | 7 Some applications then get 𝑏𝑘 − 𝑎𝑘 = 2𝜋/𝑛, and hence |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| ≤ 𝜔∞ (𝑓; 2𝜋/𝑛)

(𝑘 = 1, 2, . . . , 𝑛) ,

and so 𝑁

𝑁

𝑘=1

𝑘=1

∑ |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )|2 = ∑ 𝜆 𝑘 |𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| ≤ VarΛ (𝑓; [0, 2𝜋])

|𝑓(𝑏𝑘 ) − 𝑓(𝑎𝑘 )| 𝜆𝑘

𝜔∞ (𝑓; 2𝜋/𝑁) 𝜆𝑁

for 𝑛 ∈ ℕ since the sequence (𝜆 𝑛)𝑛 is decreasing. Consequently, 2𝜋

𝑁 ∫ |𝑓(𝑥 + 𝜋/𝑛) − 𝑓(𝑥 − 𝜋/𝑛)|2 𝑑𝑥 ≤ 2𝜋 VarΛ (𝑓; [0, 2𝜋]) 0

or

𝑁

∑ (𝛼𝑛2 + 𝛽𝑛2 )

𝑛=1

𝜔∞ (𝑓; 2𝜋/𝑁) 𝜆𝑁

𝜔 (𝑓; 2𝜋/𝑁) sin2 𝑛𝜋 ≤ VarΛ (𝑓; [0, 2𝜋]) ∞ , 2𝑁 2𝑁𝜆 𝑁

where 𝛼𝑛 and 𝛽𝑛 denote the Fourier coefficients (7.2) and (7.3) of 𝑓. Setting now 𝑁 = 2𝑚 , we obtain 2𝑚 𝜔 (𝑓; 2𝜋/2𝑚 ) ∑ (𝛼𝑛2 + 𝛽𝑛2 ) ≤ VarΛ (𝑓; [0, 2𝜋]) ∞𝑚+1 . 2 𝜆 2𝑚+1 𝑛=2𝑚−1 +1 Thus, 2𝑚

∑ 𝑛=2𝑚−1 +1

(𝛼𝑛2

+

𝛽𝑛2 )1/2

𝑚/2

≤2

2𝑚

{ ∑ 𝑛=2𝑚−1 +1

1/2

(𝛼𝑛2

≤ √VarΛ (𝑓; [0, 2𝜋])

+

𝛽𝑛2 )}

√𝜔∞ (𝑓; 2𝜋/2𝑚 ) √2𝜆 2𝑚+1

,

and so ∞

∑ √𝛼𝑛2 + 𝛽𝑛2 ≤ √

𝑛=2

VarΛ (𝑓; [0, 2𝜋]) ∞ √𝜔∞ (𝑓; 2𝜋/2𝑚 ) ∑ . 2 √𝜆 2𝑚+1 𝑚=1

(7.23)

However, the convergence of the series on the right-hand side of (7.23) is equiva­ lent to that of the series (7.22) if the terms of this series are decreasing from some point on.³ This completes the proof. For the special Waterman sequence Λ 𝑞 = (𝑛−𝑞 )𝑛 , see Definition 2.29, we obtain from Theorem 7.1 the following

3 Here, we use the condensation criterion for series with (eventually) decreasing terms.

7.2 Fourier series and Waterman spaces | 431

Corollary 7.2. For 0 < 𝑞 ≤ 1, let 𝑓 ∈ 𝐶([0, 2𝜋]) ∩ Λ 𝑞 𝐵𝑉([0, 2𝜋]) be 2𝜋-periodic. Suppose that ∞ √𝜔∞ (𝑓; 2𝜋/𝑛) ∑ < ∞, (7.24) 𝑛1−𝑞/2 𝑛=1 and the convergence in (7.24) is monotone. Then the Fourier series (7.1) of 𝑓 converges absolutely. In particular, this holds for 𝑓 ∈ 𝐶([0, 2𝜋]) ∩ 𝐻𝐵𝑉([0, 2𝜋]) if ∞

√𝜔∞ (𝑓; 2𝜋/𝑛) < ∞. √𝑛 𝑛=1 ∑

(7.25)

Now, we consider some convergence results for functions in the Waterman space Λ𝐵𝑉 in more detail. First, we recall a notion from the theory of Fourier series. Definition 7.3. Let 𝛼, 𝛽 ∈ ℝ. A real series ∞

𝛤 := ∑ 𝛾𝑛 𝑛=1

is called (𝐶, 𝛼)-summable if it converges and 𝛾𝑛 = 𝑜(𝑛−𝛼 ) as 𝑛 → ∞. Moreover, the series 𝛤 is called (𝐶, 𝛽)-bounded if its partial sums are bounded and 𝛾𝑛 = 𝑂(𝑛−𝛽 ) as 𝑛 → ∞. ◼ For example, a classical result [132] asserts that the Fourier series 𝑆[𝑓] in (7.1) of a func­ tion 𝑓 ∈ 𝐵𝑉([0, 2𝜋]) is (𝐶, −1)-bounded, convergent everywhere, and even uniformly convergent on each closed interval of continuity. It follows then from a result of Hardy and Littlewood (see, e.g. [140, p. 121]) that 𝑆[𝑓] is (𝐶, 𝛼)-summable for each 𝛼 > −1, and even uniformly (𝐶, 𝛼)-summable on each closed interval of continuity. Observe that the conclusion (7.19) of the Sahney–Waterman criterion states in this terminology that the Fourier series (7.1) is (𝐶, 𝛽)-summable. Since 𝐵𝑉 may be considered as a special case of the Waterman spaces Λ 𝑞 𝐵𝑉 (namely, for 𝑞 = 0), it is a tempting idea to try to extend this result to Λ 𝑞 𝐵𝑉, estab­ lishing some kind of “interaction” between 𝑞, 𝛼, and 𝛽 (if there is any). This has been done by Waterman [315]; his main theorem reads as follows. Theorem 7.4. Let 0 < 𝑞 < 1, and let 𝑓 ∈ Λ 𝑞 𝐵𝑉([0, 2𝜋]). Then the Fourier series (7.1) of 𝑓 is everywhere (𝐶, 𝛽)-bounded for 𝛽 = 𝑞 − 1, and (𝐶, 𝛼)-summable for 𝛼 > 𝑞 − 1. Proof. If we show that 𝑆[𝑓] is (𝐶, 𝛽)-bounded at a point, it follows from a well-known convexity theorem [140, p. 127] that 𝑆[𝑓] is (𝐶, 𝛼)-summable at that point for 𝛼 > 𝛽. So, we need only verify condition (7.17) for 𝛽 = 𝑞 − 1. For 𝜂𝑛 as in (7.18) and 𝛿 ∈ (𝜂𝑛 , 𝜋), consider 𝛿

𝐼𝑛 (𝑥, 𝛿) := ∫ 𝜂𝑛

|𝜑𝑥 (𝑡) − 𝜑𝑥 (𝑡 + 𝜂𝑛 )| 𝑑𝑡 . 𝑡𝛽+1

(7.26)

432 | 7 Some applications We estimate this integral in the form 1 + [𝐼 (𝑥, 𝛿) + 𝐼𝑛− (𝑥, 𝛿)] , 2 𝑛

𝐼𝑛 (𝑥, 𝛿) ≤ where

𝛿

𝐼𝑛+ (𝑥, 𝛿) := ∫ 𝜂𝑛

and

𝛿

𝐼𝑛− (𝑥, 𝛿)

:= ∫ 𝜂𝑛

|𝑓(𝑥 + 𝑡) − 𝑓(𝑥 + 𝑡 + 𝜂𝑛 )| 𝑑𝑡 𝑡𝛽+1

(7.27)

|𝑓(𝑥 − 𝑡) − 𝑓(𝑥 − 𝑡 − 𝜂𝑛 )| 𝑑𝑡 . 𝑡𝛽+1

(7.28)

For 𝑚𝑛 := ent(𝛿/𝜂𝑛 ), the integer part of 𝛿/𝜂𝑛 , and 𝑞 = 𝛽 + 1, we have (𝑖+1)𝜂𝑛

𝑚 𝐼𝑛+ (𝑥, 𝛿) 1 𝑛 ≤ ∑ ∫ 𝑛𝛽 𝑛𝛽 𝑖=1 𝑖𝜂𝑛

≤ ≤

1

𝑚𝑛

∑

𝛽 𝑛𝛽 𝜂𝑛 𝑖=1

|𝑓(𝑥 + 𝑡) − 𝑓(𝑥 + 𝑡 + 𝜂𝑛 )| 𝑑𝑡 𝑡𝑞

osc(𝑓; [𝑥 + 𝑖𝜂𝑛 , 𝑥 + 𝛿 + 2𝜂𝑛 ]) 𝑖𝑞

2 VarΛ 𝑞 (𝑓; [𝑥 + 𝑖𝜂𝑛 , 𝑥 + 𝛿 + 2𝜂𝑛 ]) , 𝜋𝛽

where osc(𝑓; [𝑎, 𝑏]) denotes the oscillation (1.12) of 𝑓 on [𝑎, 𝑏] and VarΛ 𝑞 (𝑓; [𝑎, 𝑏]) the Λ 𝑞 -variation of 𝑓 on [𝑎, 𝑏] defined in Definition 2.29. Applying Exercise 2.15, we see that 𝐼𝑛+ (𝑥, 𝛿) = 𝑜(1) (𝛿 → 0, 𝑛 → ∞) . (7.29) 𝑛𝛽 In the same way, one may show that 𝐼𝑛− (𝑥, 𝛿) = 𝑜(1) 𝑛𝛽

(𝛿 → 0, 𝑛 → ∞) ,

and so

𝐼𝑛 (𝑥, 𝛿) = 𝑜(1) (𝛿 → 0, 𝑛 → ∞) (7.30) 𝑛𝛽 for each 𝑥, and the convergence is uniform at each point of continuity of 𝑓. Suppose now that 𝑓 ∈ Λ𝑐 𝐵𝑉([0, 2𝜋]), i.e. 𝑓 is continuous in Λ-variation, see Defi­ nition 2.37. Then we may estimate the integral 𝜋−𝜂𝑛

𝐽𝑛 (𝑥, 𝛿) := ∫ 𝛿

in the form 𝐽𝑛 (𝑥, 𝛿) ≤

|𝜑𝑥 (𝑡) − 𝜑𝑥 (𝑡 + 𝜂𝑛 )| 𝑑𝑡 𝑡𝛽+1

1 + [𝐽 (𝑥, 𝛿) + 𝐽𝑛− (𝑥, 𝛿)] 2 𝑛

(7.31)

7.2 Fourier series and Waterman spaces |

as above, where

𝜋−𝜂𝑛

𝐽𝑛+ (𝑥, 𝛿) := ∫ 𝛿

and

𝜋−𝜂𝑛

𝐽𝑛− (𝑥, 𝛿) := ∫ 𝛿

433

|𝑓(𝑥 + 𝑡) − 𝑓(𝑥 + 𝑡 + 𝜂𝑛 )| 𝑑𝑡 𝑡𝛽+1

(7.32)

|𝑓(𝑥 − 𝑡) − 𝑓(𝑥 − 𝑡 − 𝜂𝑛 )| 𝑑𝑡 . 𝑡𝛽+1

(7.33)

For 𝑚𝑛 and 𝑞, as before, we then have 𝐽𝑛+ (𝑥, 𝛿) 1 𝑛−1 osc(𝑓; [𝑥 + 𝑖𝜂𝑛 , 𝑥 + (𝑖 + 2)𝜂𝑛 ]) ≤ ∑ 𝑖𝑞 𝑛𝛽 𝑛𝛽 𝑖=𝑚𝑛 ≤

2 VarΛ𝑚𝑞 𝑛 (𝑓; [𝑥 + 𝑖𝜂𝑛 , 𝑥 + (𝑖 + 2)𝜂𝑛 ]) , 𝜋𝛽

and similarly for 𝐽𝑛− (𝑥, 𝛿), where VarΛ𝑚𝑞 (𝑓; [𝑎, 𝑏]) denotes the shifted Λ 𝑞 -variation of 𝑓 in the sense of Definition 2.37. Our assumption 𝑓 ∈ Λ𝑐 𝐵𝑉([0, 2𝜋]) implies that VarΛ𝑚𝑞 (𝑓; [0, 2𝜋]) is dominated by VarΛ 𝑞 (𝑓; [0, 2𝜋]), and so 𝐽𝑛 (𝑥, 𝛿) = 𝑂(1) 𝑛𝛽

(7.34)

(𝑛𝛿 → ∞)

uniformly in 𝑥. Now, choose (𝛿𝑛 )𝑛 converging monotonically to zero such that 𝑛𝛿 → ∞ as 𝑛 → ∞. We may then summarize our results as follows. If 𝑓 ∈ Λ𝐵𝑉([0, 2𝜋]), then 𝐼𝑛 (𝑥, 𝛿𝑛 ) + 𝐽𝑛 (𝑥, 𝛿𝑛 ) = 𝑜(1) + 𝑂(1) = 𝑂(1) 𝑛𝛽

(𝑛𝛿 → ∞) .

(7.35)

Consequently, 𝑆[𝑓] is everywhere (𝐶, 𝛽)-bounded, and uniformly (𝐶, 𝛽)-bounded on each closed interval of continuity. On the other hand, if 𝑓 ∈ Λ𝑐 𝐵𝑉([0, 2𝜋]), then even 𝐼𝑛 (𝑥, 𝛿𝑛 ) + 𝐽𝑛 (𝑥, 𝛿𝑛 ) = 𝑜(1) (𝑛𝛿 → ∞) . (7.36) 𝑛𝛽 Consequently, in this case, 𝑆[𝑓] converges to 𝑓(𝑥) and is everywhere (𝐶, 𝛽)-summable. The proof is complete. Note that the result for 𝐵𝑉-functions mentioned before Theorem 7.4 is formally con­ tained in this theorem if we take 𝑞 = 0. The following theorem, also due to Water­ man [315], shows that Theorem 7.4 is, in a certain sense, sharp within the class of Λ𝐵𝑉-spaces: Theorem 7.5. Let 0 < 𝑞 < 1, and let Λ = (𝜆 𝑛 )𝑛 be some Waterman sequence such that Λ𝐵𝑉([0, 2𝜋]) ⊃ Λ 𝑞 𝐵𝑉([0, 2𝜋]) .

(7.37)

Then there exists a function 𝑓 ∈ 𝐶([0, 2𝜋]) ∩ Λ𝐵𝑉([0, 2𝜋]) whose Fourier series (7.1) is not (𝐶, 𝑞 − 1)-bounded at some point.

434 | 7 Some applications Proof. By our assumption (7.37), we may find a decreasing positive sequence (𝜇𝑛 )𝑛 which converges to zero and satisfies ∞

∞

∑ 𝜆 𝑛𝜇𝑛 < ∞,

𝑛=1

𝜇𝑛 = ∞. 𝑞 𝑛=1 𝑛

(7.38)

∑

For fixed 𝑛, let 𝑎𝑘,𝑛 := (2𝑘 +

𝑞−1 𝜋 ) 2 𝑛+

and 𝑏𝑘,𝑛 := (2𝑘 + 1 +

𝑞−1 𝜋 ) 2 𝑛+

4𝑘 + 𝑞 − 1 𝜋 2𝑛 + 𝑞

=

𝑞 2

=

𝑞 2

4𝑘 + 𝑞 + 1 𝜋. 2𝑛 + 𝑞

We define functions 𝑓𝑛 : [0, 2𝜋] → ℝ and 𝑓 : [0, 2𝜋] → ℝ by 𝑛−1

∞

𝑓𝑛 (𝑥) := ∑ 𝜇𝑘 𝜒𝑘 (𝑥),

𝑓(𝑥) := ∑ 𝜇𝑘 𝜒𝑘 (𝑥)

𝑘=1

(0 ≤ 𝑥 ≤ 2𝜋) ,

(7.39)

𝑘=1

where 𝜒𝑘 denotes the characteristic function of the interval [𝑎𝑘,𝑛 , 𝑏𝑘,𝑛 ], and extend 𝑓 with period 2𝜋 to the whole real line. Since the Waterman variations of 𝑓𝑛 satisfy 𝑛−1

(7.40)

VarΛ (𝑓𝑛 ; [0, 2𝜋]) ≤ 2 ∑ 𝜆 𝑛 𝜇𝑛 , 𝑘=1

they are uniformly bounded, by (7.38), and so we have 𝑓 ∈ Λ𝐵𝑉([0, 2𝜋]). On the other hand, the Fourier series (7.1) of 𝑓 is not (𝐶, 𝑞 − 1)-bounded at zero, which may be seen as follows. Let 𝜎𝑛𝑞 (𝑥) = 𝜎𝑛𝑞 (𝑥; 𝑓) denote the 𝑛-th (𝐶, 𝑞 − 1)-mean of 𝑆[𝑓] at 𝑥. Then we have 2𝜋

𝑞

𝜋𝜎𝑛𝑞 (0; 𝑓) = ∫ 𝑓𝑛 (𝑡)

sin [(𝑛 + 2 ) 𝑡 −

𝑞−1 𝜇𝑛 (2 sin(𝑡/2))2

0 𝑛−1

𝑏𝑘,𝑛

=∑ ∫

𝜋(𝑞−1) ] 2

𝑞

sin [(𝑛 + 2 ) 𝑡 −

𝑘=1 𝑎𝑘,𝑛

0

𝜋(𝑞−1) ] 2

𝑞−1 𝜇𝑛 (2 sin(𝑡/2))2

2𝜋

𝑑𝑡 + 2 ∫ 𝑓𝑛 (𝑡) 𝑛−1

𝑏𝑘,𝑛

𝑑𝑡 + 2 ∑ ∫ 𝑘=1 𝑎𝑘,𝑛

𝜃(𝑡)(𝑞 − 1) 𝑑𝑡 𝑛(sin(𝑡/2))2

𝜃(𝑡)(𝑞 − 1) 𝑑𝑡 , 𝑛(sin(𝑡/2))2

where 𝜃 is a suitable function satisfying |𝜃(𝑡)| ≤ 1 (see, e.g. [329, p. 85]). However, 𝑏𝑘,𝑛

∫ 𝑎𝑘,𝑛

and

𝑞

sin [(𝑛 + 2 ) 𝑡 −

𝜋(𝑞−1) ] 2

𝑞−1 𝜇𝑛 (2 sin(𝑡/2))2

𝑏𝑘,𝑛

∫ 𝑎𝑘,𝑛

𝑑𝑡 ≥

1 𝑞−1 𝜇𝑛

𝜇𝑘 𝜇𝑘 2 𝑞 ≥ 𝐶1 𝑞 𝑛 + 𝑞/2 𝑏𝑘,𝑛 𝑘

𝜋𝜇𝑘 𝜃(𝑡)(𝑞 − 1) 𝜇 1 𝜋 𝑑𝑡 ≤ ≤ 𝐶2 𝑘2 . 𝑛(sin(𝑡/2))2 𝑛 𝑛 + 𝑞/2 2𝑎𝑘,𝑛 𝑘

(7.41)

(7.42)

7.3 Applications to nonlinear integral equations |

435

Taking the sum over 𝑘 = 1, 2, . . . , 𝑛 − 1 and letting 𝑛 → ∞, we see that the last expression in (7.41) becomes unbounded, by (7.38), while the last expression in (7.41) remains bounded. We conclude that ‖𝜎𝑛𝑞 (0; ⋅)‖∞ → ∞ as 𝑛 → ∞, and so the sequence (𝜎𝑛𝑞 (0; 𝑓))𝑛 is unbounded for the function 𝑓 ∈ Λ𝐵𝑉([0, 2𝜋]) constructed above. Moreover, since 𝐶([0, 2𝜋]) ∩ Λ𝐵𝑉([0, 2𝜋]) is a closed subspace of Λ𝐵𝑉([0, 2𝜋]) and the functions 𝑓𝑛 in (7.39) can be modified so as to be continuous without any substan­ tial change in the argument, we have the desired result. Other convergence criteria which are formulated in terms of moduli of variation and Chanturiya classes (Definition 2.35) will be given in Section 7.4.

7.3 Applications to nonlinear integral equations In this section, we apply the abstract results of Chapter 5 and Chapter 6 to obtain exis­ tence (and sometimes uniqueness) of solutions of certain nonlinear integral equations over the interval [0, 1]. We start with an existence and uniqueness result for solutions of bounded 𝑝-variation in Wiener’s or Riesz’s sense. To get a unified approach, we use the shortcut Var𝑝 (𝑓) to denote either the Wiener 𝑝-variation Var𝑊 𝑝 (𝑓; [0, 1]) defined in (1.61) or the Riesz 𝑝-variation Var𝑅𝑝 (𝑓; [0, 1]) defined in (2.88), and 𝐵𝑉𝑝 to denote either the space 𝑊𝐵𝑉𝑝 ([0, 1]) or the space 𝑅𝐵𝑉𝑝 ([0, 1]). Consider the nonlinear integral equations of Hammerstein type 1

𝑓(𝑠) = ∫ 𝑘(𝑠, 𝑡)ℎ(𝑡, 𝑓(𝑡)) 𝑑𝑡 + 𝑏(𝑠) (0 ≤ 𝑠 ≤ 1) .

(7.43)

0

Here, 𝑘 : [0, 1] × [0, 1] → ℝ is a given kernel function (whose properties will be made precise below), and 𝑏 ∈ 𝐵𝑉𝑝 is also given. We are interested in conditions on the nonlinearity ℎ : [0, 1] × ℝ → ℝ under which (7.43) admits a (unique) solution 𝑓 ∈ 𝐵𝑉𝑝 . Denoting the corresponding linear integral operator in (7.43) by 1

𝐾𝑔(𝑠) := ∫ 𝑘(𝑠, 𝑡)𝑔(𝑡) 𝑑𝑡

(0 ≤ 𝑠 ≤ 1) ,

(7.44)

0

we may rewrite (7.43) as operator equation 𝑓 − 𝐾𝑆ℎ 𝑓 = 𝑏 ,

(7.45)

where 𝑆ℎ is the superposition operator (6.1), and try to apply the familiar fixed point principles to (7.45). To this end, we assume that the superposition operator (6.1) in­ duced by ℎ maps the space 𝐵𝑉𝑝 into itself and is bounded in the corresponding norm.⁴

4 Sufficient conditions for this in case 𝑝 = 1 have been given in Theorem 6.11.

436 | 7 Some applications Now, we impose some hypotheses on the kernel function 𝑘 in (7.43). Suppose that 𝑘(𝑠, ⋅) ∈ 𝐿 1 ([0, 1]) for 0 ≤ 𝑠 ≤ 1, and that the function 𝑣𝑝 : [0, 1] → ℝ defined by 𝑣𝑝 (𝑡) := Var𝑝 (𝑘(⋅, 𝑡))1/𝑝 belongs to 𝐿 𝑝 ([0, 1]). To simplify the notation, we use the shortcut 1

1/𝑝

1

𝜅𝑝 := ‖𝑘(0, ⋅)‖𝐿 1 + ‖𝑣𝑝 ‖𝐿 𝑝 = ∫ |𝑘(0, 𝑡)| 𝑑𝑡 + (∫ Var𝑝 (𝑘(⋅, 𝑡)) 𝑑𝑡) 0

.

0

In the following Lemma 7.6, we give a sufficient condition under which the oper­ ator 𝑓 󳨃→ 𝐾𝑆ℎ 𝑓 + 𝑏 (whose fixed points coincide with the solutions of (7.45) leaves a closed ball in the space 𝐵𝑉𝑝 ([0, 1]) invariant. In case 𝑝 = 1, i.e. in the space 𝐵𝑉([0, 1]), similar conditions of this type have been considered in [67]. Lemma 7.6. Under the above hypotheses, let 𝑟 > 0 be so large that ̃ < 𝑟, 𝜅𝑝 𝑘(𝑟)

(7.46)

̃ := sup {|ℎ(𝑡, 𝑢)| : 0 ≤ 𝑡 ≤ 1, |𝑢| ≤ 𝑟} . 𝑘(𝑟)

(7.47)

̃ is given by where 𝑘(𝑟)

Then the operator 𝑓 󳨃→ 𝐾𝑆ℎ 𝑓 + 𝑏 maps, for any 𝑏 ∈ 𝐵𝑉𝑝 satisfying ̃ , ‖𝑏‖𝐵𝑉𝑝 ≤ 𝑟 − 𝜅𝑝 𝑘(𝑟)

(7.48)

the closed ball 𝐵𝑟 (𝐵𝑉𝑝 ) = {𝑓 ∈ 𝐵𝑉𝑝 : ‖𝑓‖𝐵𝑉𝑝 ≤ 𝑟} into itself. Proof. For the sake of definiteness, let us prove the assertion in the space 𝑊𝐵𝑉 𝑝 ([0, 1]). Suppose that 𝑟 satisfies (7.46), and 𝑏 satisfies (7.48). Given 𝑓 ∈ 𝐵𝑟 (𝑊𝐵𝑉𝑝 ) and a parti­ tion {𝑠0 , 𝑠1 , . . . , 𝑠𝑚 } ∈ P([0, 1]), we get⁵ 󵄨󵄨𝑝 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∑ |𝐾𝑆ℎ 𝑓(𝑠𝑗 ) − 𝐾𝑆ℎ 𝑓(𝑠𝑗−1 )| = ∑ 󵄨󵄨∫ [𝑘(𝑠𝑗 , 𝑡) − 𝑘(𝑠𝑗−1 , 𝑡)] ℎ(𝑡, 𝑓(𝑡)) 𝑑𝑡󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨 𝑗=1 𝑗=1 󵄨 0 󵄨󵄨 󵄨 𝑚

𝑝

𝑚

1 𝑝

𝑚

≤ sup |ℎ(𝑡, 𝑓(𝑡))| ∫ ∑ |𝑘(𝑠𝑗 , 𝑡) − 𝑘(𝑠𝑗−1 , 𝑡)|𝑝 𝑑𝑡 0≤𝑡≤1

0 𝑗=1

1

̃ 𝑝 ∫ 𝑣 (𝑡)𝑝 𝑑𝑡 , ≤ 𝑘(𝑟) 𝑝 0

5 Here, we use the fact that 𝑊𝐵𝑉𝑝 󳨅→ 𝐵 with imbedding constant 𝑐(𝑊𝐵𝑉𝑝 , 𝐵) = 1, see (0.36), and hence ‖𝑓‖∞ ≤ 𝑟 for 𝑓 ∈ 𝐵𝑟 (𝑊𝐵𝑉𝑝 ).

7.3 Applications to nonlinear integral equations

| 437

and so, passing to the supremum over all partitions {𝑠0 , 𝑠1 , . . . , 𝑠𝑚 } ∈ P([0, 1]), 𝑝

̃ 𝑝 Var𝑊 𝑝 (𝐾𝑆ℎ 𝑓; [0, 1]) ≤ 𝑘(𝑟) ‖𝑣𝑝 ‖𝐿 𝑝 . Consequently, from (7.48), we conclude that 1/𝑝 + ‖𝑏‖𝑊𝐵𝑉𝑝 ‖𝐾𝑆ℎ 𝑓 + 𝑏‖𝑊𝐵𝑉𝑝 ≤ |𝐾𝑆ℎ 𝑓(0)| + Var𝑊 𝑝 (𝐾𝑆ℎ 𝑓; [0, 1]) 1

̃ ≤ ∫ |𝑘(0, 𝑡)ℎ(𝑡, 𝑓(𝑡))| 𝑑𝑡 + 𝑘(𝑟)‖𝑣 𝑝 ‖𝐿 𝑝 + ‖𝑏‖𝑊𝐵𝑉𝑝 0

̃ ̃ ̃ ≤ 𝑘(𝑟)‖𝑘(0, ⋅)‖𝐿 1 + 𝑘(𝑟)‖𝑣 𝑝 ‖𝐿 𝑝 + ‖𝑏‖𝑊𝐵𝑉𝑝 ≤ 𝑘(𝑟)𝜅𝑝 + ‖𝑏‖𝑊𝐵𝑉𝑝 ≤ 𝑟 , which proves the assertion for 𝑊𝐵𝑉𝑝 . The proof for 𝑅𝐵𝑉𝑝 ([0, 1]) is exactly the same, with obvious modifications in the definition of variations. In the following Lemma 7.7, we give a sufficient condition under which the operator 𝑓 󳨃→ 𝐾𝑆ℎ 𝑓 + 𝑏 is a contraction in the norm (1.65) (for 𝑊𝐵𝑉𝑝 ([0, 1])), respectively (2.90) (for 𝑅𝐵𝑉𝑝 ([0, 1])). Lemma 7.7. Under the above hypotheses, let 𝑟 > 0 be so small that 𝜅𝑝 𝑘(𝑟) < 1 ,

(7.49)

where 𝑘(𝑟) is given by (6.10). Then the operator 𝑓 󳨃→ 𝐾𝑆ℎ 𝑓 + 𝑏 is, for any 𝑏 ∈ 𝐵𝑉𝑝 , a contraction on 𝐵𝑟 (𝐵𝑉𝑝 ) with respect to the norm (1.65), respectively (2.90). Proof. We work again in the space 𝑊𝐵𝑉𝑝 ([0, 1]). Suppose that 𝑟 satisfies (7.49), and let 𝑓, 𝑔 ∈ 𝐵𝑟 (𝑊𝐵𝑉𝑝 ) be fixed. We claim that ‖𝐾𝑆ℎ 𝑓 − 𝐾𝑆ℎ 𝑔‖𝑊𝐵𝑉𝑝 ≤ 𝜅𝑝 𝑘(𝑟)‖𝑓 − 𝑔‖𝑊𝐵𝑉𝑝 , which, together with (7.49), proves the assertion. First of all, we have 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 |𝐾𝑆ℎ 𝑓(0) − 𝐾𝑆ℎ 𝑔(0)| = 󵄨󵄨∫ 𝑘(0, 𝑡)[ℎ(𝑡, 𝑓(𝑡)) − ℎ(𝑡, 𝑔(𝑡))] 𝑑𝑡󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 󵄨󵄨 1

(7.50)

≤ 𝑘(𝑟) ∫ |𝑘(0, 𝑡)| |𝑓(𝑡) − 𝑔(𝑡)| 𝑑𝑡 ≤ 𝑘(𝑟)‖𝑘(0, ⋅)‖𝐿 1 ‖𝑓 − 𝑔‖∞ 0

≤ 𝑘(𝑟)‖𝑘(0, ⋅)‖𝐿 1 ‖𝑓 − 𝑔‖𝑊𝐵𝑉𝑝 , where again we have used the fact that 𝑊𝐵𝑉𝑝 ([0, 1]) is continuously imbedded into the space of all bounded functions on [0, 1] with the supremum norm (0.39). On the other hand, we show now that also 1/𝑝 Var𝑊 ≤ 𝑘(𝑟)‖𝑣𝑝 ‖𝐿 𝑝 ‖𝑓 − 𝑔‖𝑊𝐵𝑉𝑝 . 𝑝 (𝐾𝑆ℎ 𝑓 − 𝐾𝑆ℎ 𝑔; [0, 1])

438 | 7 Some applications For any partition {𝑠0 , 𝑠1 , . . . , 𝑠𝑚 } ∈ P([0, 1]), we have 𝑚

∑ |𝐾𝑆ℎ 𝑓(𝑠𝑗 ) − 𝐾𝑆ℎ 𝑓(𝑠𝑗−1 ) − 𝐾𝑆ℎ 𝑔(𝑠𝑗 ) + 𝐾𝑆ℎ 𝑔(𝑠𝑗−1 )|𝑝

𝑗=1

󵄨󵄨 1 󵄨󵄨𝑝 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 = ∑ 󵄨󵄨∫[𝑘(𝑠𝑗 , 𝑡) − 𝑘(𝑠𝑗−1 , 𝑡)][𝑆ℎ 𝑓(𝑡) − 𝑆ℎ 𝑔(𝑡)] 𝑑𝑡󵄨󵄨󵄨 󵄨 󵄨󵄨 󵄨 𝑗=1 󵄨 0 󵄨󵄨 󵄨 𝑚

𝑚

1

≤ ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖𝑝∞ ∑ ∫ |𝑘(𝑠𝑗 , 𝑡) − 𝑘(𝑠𝑗−1 , 𝑡)|𝑝 𝑑𝑡 𝑗=1 0

1

≤ ‖𝑆ℎ 𝑓 −

𝑝 𝑆ℎ 𝑔‖𝑊𝐵𝑉𝑝

𝑚

∫ ∑ |𝑘(𝑠𝑗 , 𝑡) − 𝑘(𝑠𝑗−1 , 𝑡)|𝑝 𝑑𝑡 0 𝑗=1 1

𝑝

𝑝

𝑝

≤ 𝑘(𝑟)𝑝 ‖𝑓 − 𝑔‖𝑊𝐵𝑉𝑝 ∫ 𝑣𝑝 (𝑡)𝑝 𝑑𝑡 = 𝑘(𝑟)𝑝 ‖𝑣𝑝 ‖𝐿 𝑝 ‖𝑓 − 𝑔‖𝑊𝐵𝑉𝑝 . 0

Passing again to the supremum with respect to all partitions {𝑠0 , 𝑠1 , . . . , 𝑠𝑚 } ∈ P([0, 1]), we obtain 𝑝

𝑝

𝑝 Var𝑊 𝑝 (𝐾𝑆ℎ 𝑓 − 𝐾𝑆ℎ 𝑔; [0, 1]) ≤ 𝑘(𝑟) ‖𝑣𝑝 ‖𝐿 𝑝 ‖𝑓 − 𝑔‖𝑊𝐵𝑉𝑝 ,

and combining this with (7.50) yields 1/𝑝 ‖𝐾𝑆ℎ 𝑓 − 𝐾𝑆ℎ 𝑔‖𝑊𝐵𝑉𝑝 = |𝐾𝑆ℎ 𝑓(0) − 𝐾𝑆ℎ 𝑔(0)| + Var𝑊 𝑝 (𝐾𝑆ℎ 𝑓 − 𝐾𝑆ℎ 𝑔; [0, 1])

≤ 𝑘(𝑟)‖𝑘(0, ⋅)‖𝐿 1 ‖𝑓 − 𝑔‖𝑊𝐵𝑉𝑝 + 𝑘(𝑟)‖𝑣𝑝 ‖𝐿 𝑝 ‖𝑓 − 𝑔‖𝑊𝐵𝑉𝑝 = 𝑘(𝑟)𝜅𝑝 ‖𝑓 − 𝑔‖𝑊𝐵𝑉𝑝 , and establishes the result. The proof for 𝑅𝐵𝑉𝑝 is similar. Combining Theorem 6.13 (for 𝜙(𝑡) = |𝑡|𝑝 ), Lemma 7.6, and Lemma 7.7, and observing that the fixed points of the operator 𝑓 󳨃→ 𝐾𝑆ℎ 𝑓 + 𝑏 coincide with the solutions of (7.45), Banach’s contraction mapping theorem now implies the following Theorem 7.8. Suppose that there exists 𝑟 > 0 such that both estimates (7.46) and (7.49) hold. Then the nonlinear integral equation (7.43) has, for each 𝑏 ∈ 𝐵𝑉𝑝 satisfying (7.48), a unique solution 𝑓 ∈ 𝐵𝑉𝑝 . Let us return for a moment to the autonomous case of the operator (5.1) where, as one could expect, our calculations simplify.⁶ We suppose, in addition, that the kernel func­ tion splits into two functions of one variable, i.e. 𝑘(𝑠, 𝑡) = 𝑙(𝑠)𝑚(𝑡), where 𝑙 ∈ 𝐵𝑉𝑝 ([0, 1]) and 𝑚 ∈ 𝐿 𝑝 ([0, 1]). In this case, we get 𝑣𝑝 (𝑡) = Var𝑝 (𝑙)1/𝑝 |𝑚(𝑡)| ,

6 For instance, the characteristic (7.47) takes the simpler form (5.15).

7.3 Applications to nonlinear integral equations |

439

and hence 1

1/𝑝

1 𝑝

𝜅𝑝 = ∫ |𝑙(0)| |𝑚(𝑡)| 𝑑𝑡 + (∫ Var𝑝 (𝑙)|𝑚(𝑡)| 𝑑𝑡) 0

0 1/𝑝

= |𝑙(0)| ‖𝑚‖𝐿 1 + Var𝑝 (𝑙)

‖𝑚‖𝐿 𝑝 ≤ ‖𝑙‖𝐵𝑉𝑝 ‖𝑚‖𝐿 1 .

Therefore, the crucial condition (7.46) in Lemma 7.6 holds if ̃ < 𝑟, ‖𝑙‖𝐵𝑉𝑝 ‖𝑚‖𝐿 1 𝑘(𝑟)

(7.51)

̃ given by (5.15), while the crucial condition (7.49) in Lemma 7.7 holds if with 𝑘(𝑟) (7.52)

‖𝑙‖𝐵𝑉𝑝 ‖𝑚‖𝐿 1 𝑘(𝑟) < 1

with 𝑘(𝑟) given by (5.13). We illustrate these conditions by a very elementary example. Example 7.9. First, let ℎ(𝑢) = 𝑢𝛼 , where 𝛼 ∈ ℝ, 𝛼 ≥ 2. The mean value theorem shows that ̃ = 𝑟𝛼 , 𝑘 (𝑟) = 𝛼(𝛼 − 1)𝑟𝛼−2 , 𝑘̃ (𝑟) = 𝛼𝑟𝛼−1 , 𝑘(𝑟) = 𝛼𝑟𝛼−1 , 𝑘(𝑟) 1 1 where 𝑘1 (𝑟) is given by (5.14) and 𝑘̃ 1 (𝑟) by (5.16). Thus, the estimate (7.52) reads 𝐾(𝑟) ≤ ‖𝑙‖𝐵𝑉𝑝 ‖𝑚‖𝐿 1 𝛼𝑟𝛼−1 < 1 ,

(7.53)

while the general estimate (5.82) from Section 5.5 becomes 𝐾(𝑟) ≤ ‖𝑙‖𝐵𝑉𝑝 ‖𝑚‖𝐿 1 max {𝛼(𝛼 − 1)𝑟𝛼−2 , 𝛼𝑟𝛼−1 } < 1 .

(7.54)

Of course, both estimates may be achieved for 𝑟 sufficiently small, but (7.53) is better than (7.54) for 𝑟 < 𝛼 − 1, while (7.54) is better than (7.53) for 𝑟 > 𝛼 − 1. Next, let ℎ(𝑢) = log(1 + 𝑢) for 𝑢 > 0 and ℎ(𝑢) = 0 for 𝑢 ≤ 0. Again, the mean value theorem implies that 𝑘(𝑟) ≡ 1 ,

̃ = log(1 + 𝑟) , 𝑘(𝑟)

𝑘1 (𝑟) =

1 , (1 + 𝑟)2

𝑘̃ 1 (𝑟) =

1 . 1+𝑟

So, the estimate (7.52) reads 𝐾(𝑟) ≤ ‖𝑙‖𝐵𝑉𝑝 ‖𝑚‖𝐿 1 < 1 ,

(7.55)

while the estimate (7.54) becomes 𝐾(𝑟) ≤ ‖𝑙‖𝐵𝑉𝑝 ‖𝑚‖𝐿 1 max {

‖𝑙‖𝐵𝑉𝑝 ‖𝑚‖𝐿 1 𝑟 1 } = < 1. , (1 + 𝑟)2 1 + 𝑟 1+𝑟

(7.56)

The estimate (7.55) does not depend on 𝑟, but the estimate (7.56) may be achieved for 𝑟 sufficiently large; so, in this example, (7.56) is always better than (7.55). ♥

440 | 7 Some applications Let us now prove a parallel result in the Waterman space Λ𝐵𝑉([𝑎, 𝑏]) which we in­ troduced in Definition 2.15. Since the space 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) is a proper subspace of Λ𝐵𝑉([𝑎, 𝑏]) for a suitable Waterman sequence Λ = (𝜆 𝑛 )𝑛 , the results which follow extend those given above.⁷ Consider again (7.43), where now 𝑏 ∈ Λ𝐵𝑉([0, 1]) is given, and the hypothetical so­ lution 𝑓 is searched in the same space Λ𝐵𝑉([0, 1]). As before, we write (7.43) as equiv­ alent operator equation (7.45), where 𝑆ℎ is the superposition operator (6.1) and 𝐾 is the integral operator (7.44). We impose the following conditions on the functions ℎ and 𝑘 in (7.43). Suppose that the superposition operator (6.1) induced by ℎ maps Λ𝐵𝑉([0, 1]) into itself and is bounded. Concerning the kernel function 𝑘 in (7.43), we assume as before that 𝑘(𝑠, ⋅) ∈ 𝐿 1 ([0, 1]) for 0 ≤ 𝑠 ≤ 1, and that the function 𝑣Λ : [0, 1] → ℝ defined by 𝑣Λ (𝑡) := VarΛ (𝑘(⋅, 𝑡)) belongs to 𝐿 1 ([0, 1]). To simplify the notation, we use the shortcut 1

𝜅Λ := ‖𝑘(0, ⋅)‖𝐿 1 + ‖𝑣Λ ‖𝐿 1 = ∫ [|𝑘(0, 𝑡)| + VarΛ (𝑘(⋅, 𝑡))] 𝑑𝑡 . 0

The following two lemmas are parallel to Lemma 7.6 and Lemma 7.7. Lemma 7.10. Under the above hypotheses, let 𝑟 > 0 be so large that ̃ < 𝑟, 𝜅Λ 𝑘(𝑟)

(7.57)

̃ is given by (7.47). Then the operator 𝑓 󳨃→ 𝐾𝑆 𝑓 + 𝑏 maps, for any 𝑏 ∈ Λ𝐵𝑉 where 𝑘(𝑟) ℎ satisfying ̃ , ‖𝑏‖Λ𝐵𝑉 ≤ 𝑟 − 𝜅Λ 𝑘(𝑟) (7.58) the closed ball 𝐵𝑟 (Λ𝐵𝑉) = {𝑓 ∈ Λ𝐵𝑉 : ‖𝑓‖Λ𝐵𝑉 ≤ 𝑟} into itself. Proof. Suppose that 𝑟 satisfies (7.57), and 𝑏 satisfies (7.58). Given 𝑓 ∈ 𝐵𝑟 (Λ𝐵𝑉) and an infinite collection 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([0, 1]), we get 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∑ 𝜆 𝑘 |𝐾𝑆ℎ 𝑓(𝑏𝑘 ) − 𝐾𝑆ℎ 𝑓(𝑎𝑘 )| = ∑ 𝜆 𝑘 󵄨󵄨∫ [𝑘(𝑏𝑘 , 𝑡) − 𝑘(𝑎𝑘 , 𝑡)] ℎ(𝑡, 𝑓(𝑡)) 𝑑𝑡󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨 𝑘=1 𝑘=1 󵄨󵄨 󵄨󵄨 0 ∞

∞

1

∞

≤ sup |ℎ(𝑡, 𝑓(𝑡))| ∫ ∑ 𝜆 𝑘 |𝑘(𝑏𝑘 , 𝑡) − 𝑘(𝑎𝑘 , 𝑡)| 𝑑𝑡 0≤𝑡≤1

0 𝑘=1 1

̃ ∫ 𝑣 (𝑡) 𝑑𝑡 , ≤ 𝑘(𝑟) Λ 0

7 In fact, as we have shown in Proposition 2.32, the imbedding 𝑊𝐵𝑉𝑝 󳨅→ Λ 𝑞 holds for 𝑝󸀠 𝑞 > 1.

7.3 Applications to nonlinear integral equations | 441

and so, passing to the supremum over all collections 𝑆∞ ∈ 𝛴∞ ([0, 1]), ̃ VarΛ (𝐾𝑆ℎ 𝑓; [0, 1]) ≤ 𝑘(𝑟)‖𝑣 Λ ‖𝐿 1 . Consequently, from (7.58), we conclude that ‖𝐾𝑆ℎ 𝑓 + 𝑏‖Λ𝐵𝑉 ≤ |𝐾𝑆ℎ 𝑓(0)| + VarΛ (𝐾𝑆ℎ 𝑓; [0, 1]) + ‖𝑏‖Λ𝐵𝑉 1

̃ ≤ ∫ |𝑘(0, 𝑡)ℎ(𝑡, 𝑓(𝑡))| 𝑑𝑡 + 𝑘(𝑟)‖𝑣 Λ ‖𝐿 1 + ‖𝑏‖Λ𝐵𝑉 0

̃ ̃ ̃ ≤ 𝑘(𝑟)‖𝑘(0, ⋅)‖𝐿 1 + 𝑘(𝑟)‖𝑣 Λ ‖𝐿 1 + ‖𝑏‖Λ𝐵𝑉 ≤ 𝑘(𝑟)𝜅Λ + ‖𝑏‖Λ𝐵𝑉 ≤ 𝑟 , which proves the assertion. Lemma 7.11. Under the above hypotheses, let 𝑟 > 0 be so small that 𝜅Λ max {𝑐(Λ𝐵𝑉, 𝐵), 𝑘(𝑟)} < 1 ,

(7.59)

where 𝑐(Λ𝐵𝑉, 𝐵) is the imbedding constant (0.36) of the imbedding Λ𝐵𝑉([0, 1]) 󳨅→ 𝐵([0, 1]), and 𝑘(𝑟) is given by (6.10). Then the operator 𝑓 󳨃→ 𝐾𝑆ℎ 𝑓 + 𝑏 is, for any 𝑏 ∈ Λ𝐵𝑉, a contraction on 𝐵𝑟 (Λ𝐵𝑉) with respect to the norm (2.30). Proof. Suppose that 𝑟 satisfies (7.59), and let 𝑓, 𝑔 ∈ 𝐵𝑟 (Λ𝐵𝑉) be fixed. We claim that ‖𝐾𝑆ℎ 𝑓 − 𝐾𝑆ℎ 𝑔‖Λ𝐵𝑉 ≤ 𝜅Λ max {𝑐(Λ𝐵𝑉, 𝐵), 𝑘(𝑟)}‖𝑓 − 𝑔‖Λ𝐵𝑉 , which together with (7.59) proves the assertion. First of all, we have 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 |𝐾𝑆ℎ 𝑓(0) − 𝐾𝑆ℎ 𝑔(0)| = 󵄨󵄨∫ 𝑘(0, 𝑡)[ℎ(𝑡, 𝑓(𝑡)) − ℎ(𝑡, 𝑔(𝑡))] 𝑑𝑡󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 󵄨󵄨 1

(7.60)

≤ 𝑘(𝑟) ∫ |𝑘(0, 𝑡)| |𝑓(𝑡) − 𝑔(𝑡)| 𝑑𝑡 ≤ 𝑘(𝑟)‖𝑘(0, ⋅)‖𝐿 1 ‖𝑓 − 𝑔‖∞ 0

≤ 𝑐(Λ𝐵𝑉, 𝐵)𝑘(𝑟)‖𝑘(0, ⋅)‖𝐿 1 ‖𝑓 − 𝑔‖Λ𝐵𝑉 , where we have used the fact that Λ𝐵𝑉([0, 1]) is continuously imbedded⁸ into the space of all bounded functions on [0, 1] with the supremum norm (0.39). On the other hand, we now show that also VarΛ (𝐾𝑆ℎ 𝑓 − 𝐾𝑆ℎ 𝑔; [0, 1]) ≤ 𝑐(Λ𝐵𝑉, 𝐵)‖𝑣Λ ‖𝐿 1 𝑘(𝑟)‖𝑓 − 𝑔‖Λ𝐵𝑉 .

8 Recall that 𝑐(Λ𝐵𝑉, 𝐵) ≤ max {1/𝜆 1 , 1}, see Exercise 2.17.

442 | 7 Some applications For any collection 𝑆∞ = {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([0, 1]), we have ∞

∑ 𝜆 𝑘 |𝐾𝑆ℎ 𝑓(𝑏𝑘 ) − 𝐾𝑆ℎ 𝑓(𝑎𝑘 ) − 𝐾𝑆ℎ 𝑔(𝑏𝑘 ) + 𝐾𝑆ℎ 𝑔(𝑎𝑘 )| 𝑘=1

󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 = ∑ 𝜆 𝑘 󵄨󵄨∫[𝑘(𝑏𝑘 , 𝑡) − 𝑘(𝑎𝑘 , 𝑡)][𝑆ℎ 𝑓(𝑡) − 𝑆ℎ 𝑔(𝑡)] 𝑑𝑡󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑘=1 󵄨󵄨 󵄨󵄨 0 ∞

1

∞

≤ ‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖∞ ∑ 𝜆 𝑘 ∫ |𝑘(𝑏𝑘 , 𝑡) − 𝑘(𝑎𝑘 , 𝑡)| 𝑑𝑡 𝑘=1

0 1

∞

≤ 𝑐(Λ𝐵𝑉, 𝐵)‖𝑆ℎ 𝑓 − 𝑆ℎ 𝑔‖Λ𝐵𝑉 ∫ ∑ 𝜆 𝑘 |𝑘(𝑏𝑘 , 𝑡) − 𝑘(𝑎𝑘 , 𝑡)| 𝑑𝑡 0 𝑘=1 1

≤ 𝑐(Λ𝐵𝑉, 𝐵)𝑘(𝑟)‖𝑓 − 𝑔‖Λ𝐵𝑉 ∫ 𝑣Λ (𝑡) 𝑑𝑡 = 𝑐(Λ𝐵𝑉, 𝐵)𝑘(𝑟)‖𝑣Λ ‖𝐿 1 ‖𝑓 − 𝑔‖Λ𝐵𝑉 . 0

Passing again to the supremum with respect to all infinite collections 𝑆∞ ∈ 𝛴∞ ([0, 1]), we obtain VarΛ (𝐾𝑆ℎ 𝑓 − 𝐾𝑆ℎ 𝑔; [0, 1]) ≤ 𝑐(Λ𝐵𝑉, 𝐵)𝑘(𝑟)‖𝑣Λ ‖𝐿 1 ‖𝑓 − 𝑔‖Λ𝐵𝑉 , and combining this with (7.60) yields ‖𝐾𝑆ℎ 𝑓 − 𝐾𝑆ℎ 𝑔‖Λ𝐵𝑉 = |𝐾𝑆ℎ 𝑓(0) − 𝐾𝑆ℎ 𝑔(0)| + VarΛ (𝐾𝑆ℎ 𝑓 − 𝐾𝑆ℎ 𝑔; [0, 1]) ≤ 𝑘(𝑟)‖𝑘(0, ⋅)‖𝐿 1 ‖𝑓 − 𝑔‖Λ𝐵𝑉 + 𝑐(Λ𝐵𝑉, 𝐵)𝑘(𝑟)‖𝑣Λ ‖𝐿 1 ‖𝑓 − 𝑔‖Λ𝐵𝑉 ≤ 𝜅Λ max {𝑐(Λ𝐵𝑉, 𝐵), 𝑘(𝑟)}‖𝑓 − 𝑔‖Λ𝐵𝑉 , and establishes the result. Combining Lemma 7.10 and Lemma 7.11, and observing that the fixed points of the operator 𝑓 󳨃→ 𝐾𝑆ℎ 𝑓 + 𝑏 coincide with the solutions of (7.45), Banach’s contraction mapping theorem now implies the following Theorem 7.12. Suppose that there exists 𝑟 > 0 such that both estimates (7.57) and (7.59) hold. Then the nonlinear integral equation (7.43) has, for each 𝑏 ∈ Λ𝐵𝑉 satisfying (7.58), a unique solution 𝑓 ∈ Λ𝐵𝑉. Now, we apply our abstract results to a class of singular integral equations. Consider the nonlinear weakly singular Abel–Volterra equation 𝑠

𝑓(𝑠) − ∫ 0

𝑘(𝑠, 𝑡)ℎ(𝑓(𝑡)) 𝑑𝑡 = 𝑏(𝑠) |𝑠 − 𝑡|𝜈

(0 ≤ 𝑠 ≤ 1) ,

(7.61)

where 𝑘 : [0, 1] × [0, 1] → ℝ is continuous and 0 < 𝜈 < 1. We rewrite (7.61) again as operator equation 𝑓 − 𝐾𝜈 𝐶ℎ 𝑓 = 𝑏 (𝑓 ∈ 𝑋) , (7.62)

7.3 Applications to nonlinear integral equations |

443

where 𝐾𝜈 is the weakly singular linear integral operator defined by 𝑠

𝐾𝜈 𝑔(𝑠) = ∫ 0

𝑘(𝑠, 𝑡)𝑔(𝑡) 𝑑𝑡 |𝑠 − 𝑡|𝜈

(0 ≤ 𝑠 ≤ 1) ,

(7.63)

and 𝐶ℎ is the composition operator (5.1). A very suitable space for treating such an operator equation is the Hölder space 𝐿𝑖𝑝𝑜𝛼 ([0, 1]) of all functions 𝑓 ∈ 𝐿𝑖𝑝𝛼 ([0, 1]) sat­ isfying 𝑓(0) = 0. The so-called second Hardy–Littlewood theorem (see, e.g. [19]) states that the operator (7.63) maps 𝐿𝑖𝑝𝑜𝛼 ([0, 1]) ∩ 𝐿 𝑝 ([0, 1]) into 𝐿𝑖𝑝𝑜𝛼 [0, 1] and is bounded if 1 < 𝑝 ≤ ∞, 1−𝜈

0 0 is so large that ‖𝐾𝜈 ‖𝛼 𝜇𝑟 (ℎ, 𝐿𝑖𝑝𝛼 ) < 𝑟 ,

(7.64)

where 𝜇𝑟 (ℎ, 𝐿𝑖𝑝𝛼 ) is given by Exercise 5.11, then the operator 𝑓 󳨃→ 𝐾𝜈 𝐶ℎ 𝑓 + 𝑏 maps, for any 𝑏 ∈ 𝐿𝑖𝑝𝛼 ([0, 1]) satisfying ‖𝑏‖𝐿𝑖𝑝𝛼 ≤ 𝑟 − ‖𝐾𝜈 ‖𝛼 𝜇𝑟 (ℎ, 𝐿𝑖𝑝𝛼 ) ,

(7.65)

the closed ball 𝐵𝑟 (𝐿𝑖𝑝𝛼 ) = {𝑓 ∈ 𝐿𝑖𝑝𝛼 : ‖𝑓‖𝐿𝑖𝑝𝛼 ≤ 𝑟} into itself. Proof. The sufficiency of (5.13) for the boundedness of the operator (5.1) in 𝐿𝑖𝑝𝛼 ([0, 1]) has already been shown in Theorem 5.24. Moreover, our assumption ℎ(0) = 0 guaran­ tees that the operator (5.1) maps 𝐿𝑖𝑝𝑜𝛼 ([0, 1]) into itself. Fix 𝑓 ∈ 𝐿𝑖𝑝𝑜𝛼 ([0, 1]) with ‖𝑓‖𝐿𝑖𝑝𝛼 ≤ 𝑟, where 𝑟 satisfies (7.64), and suppose that 𝑏 ∈ 𝐿𝑖𝑝𝑜𝛼 ([0, 1]) satisfies (7.65). Then ‖𝐾𝜈 𝐶ℎ 𝑓 + 𝑏‖ ≤ ‖𝐾𝜈 ‖𝛼 𝜇𝑟 (ℎ, 𝐿𝑖𝑝𝛼 ) + ‖𝑏‖𝐿𝑖𝑝𝛼 ≤ 𝑟 , by (7.65), which proves the assertion.

444 | 7 Some applications Lemma 7.14. Suppose that the derivative ℎ󸀠 of ℎ exists and satisfies (5.14). Moreover, let 𝑟 > 0 be so small that 𝑘1 (𝑟)
1, and this class of functions is preserved by a homeomorphic change of variables, as Proposition 1.12 shows.⁹ In the paper [78], the authors consider the class 𝐺𝑊([0, 2𝜋]) (respectively the class 𝑈𝐺𝑊([0, 2𝜋])) of all regular functions which have a convergent (respectively uniformly convergent) Fourier series after any change of variable. It is known that 𝐻𝐵𝑉([0, 2𝜋]) ⊆ 𝐺𝑊([0, 2𝜋]) and 𝐻𝐵𝑉([0, 2𝜋]) ∩ 𝐶([0, 2𝜋]) ⊆ 𝑈𝐺𝑊([0, 2𝜋]) . In [78], the authors also show that in the terminology of (5.3), we have the equal­ ities 𝐶𝑂𝑃(𝐺𝑊) = 𝐶𝑂𝑃(𝑈𝐺𝑊) = 𝐿𝑖𝑝𝑙𝑜𝑐 (ℝ) , which means that 𝐶ℎ (𝐺𝑊) ⊆ 𝐺𝑊 and 𝐶ℎ (𝑈𝐺𝑊) ⊆ 𝑈𝐺𝑊 if and only if ℎ is locally Lipschitz; this is perfectly analogous to Theorem 5.10. As the fundamental Theorems 7.4 and 7.5 show, the Waterman spaces Λ 𝑞 𝐵𝑉 (in particular, the space 𝐻𝐵𝑉 = Λ 1 𝐵𝑉) are most suitable for studying the convergence problems of Fourier series. For instance, the Fourier series of functions in 𝐻𝐵𝑉 con­ verge everywhere pointwise, and converge uniformly on closed intervals of continuity. This result is best possible in the sense that each larger space (within the class of Wa­ terman spaces, see Theorem 7.5 or [314]) contains a continuous function whose Fourier series diverges at some point. The functions in the Wiener–Young space 𝑊𝐵𝑉 𝜙 with complementary Young func­ tion satisfying (7.15) are contained in 𝐻𝐵𝑉, as are the functions with logarithmically integrable Banach indicatrix, see (7.16). However, one may show [314] that all these criteria are contained in the Lebesgue test. Condition (7.15) is also sharp in the sense that if ∞ 1 ∑ 𝜙∗ ( ) = ∞ , 𝑛 𝑛=1 then there is a continuous function in 𝑊𝐵𝑉𝜙 ([0, 2𝜋]) whose Fourier series diverges at some point. This was shown by Baernstein [38] and, independently, by Oskolkov [239], and answers a question raised in [132]. Needless to say, the results for Waterman spaces, which are particularly useful and natural, have been extended to more general classes like the space Λ𝐵𝑉𝜙 ([0, 2𝜋]) of functions of bounded (𝜙, Λ)-variation introduced in Definition 2.84. Recall that given a Young function 𝜙 and a Waterman sequence Λ = (𝜆 𝑛 )𝑛 , we defined the

9 In Proposition 1.12, we have proved this only for 𝑝 = 1, but the proof carries over without changes to the case 𝑝 > 1, see [225].

446 | 7 Some applications (𝜙, Λ)-variation of 𝑓 on [𝑎, 𝑏] by ∞

Var𝜙,Λ (𝑓; [𝑎, 𝑏]) = sup ∑ 𝜆 𝑛𝜙(|𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|) ,

(7.67)

𝑛=1

where the supremum is taken over all collections {[𝑎𝑛 , 𝑏𝑛 ] : 𝑛 ∈ ℕ} ∈ 𝛴∞ ([𝑎, 𝑏]). In particular, 𝑓 ∈ Λ𝐵𝑉𝑝 ([𝑎, 𝑏]) for 1 ≤ 𝑝 < ∞ if ∞

Var𝑝,Λ (𝑓; [𝑎, 𝑏]) = sup ∑ 𝜆 𝑛 |𝑓(𝑏𝑛 ) − 𝑓(𝑎𝑛 )|𝑝 < ∞ .

(7.68)

𝑛=1

The following was proved by Shiba [288]: Theorem 7.16. Let 1 < 𝑞 < ∞, 𝑞󸀠 = 𝑞/(𝑞 − 1), and 1 ≤ 𝑝 < 2𝑞. Let 𝑓 ∈ Λ𝐵𝑉𝑝 ([0, 2𝜋]) be 2𝜋-periodic. Suppose that ∞

∑

𝜔𝑝+(2−𝑝)𝑞󸀠 (𝑓; 2𝜋/𝑛)1−𝑝/2𝑞 󸀠

1/2𝑞

𝑛1−1/2𝑞 𝜆 𝑛

𝑛=1

< ∞,

(7.69)

where 𝜔𝑝 (𝑓; ⋅) denotes the 𝑝-modulus of continuity (0.98) of 𝑓 over [𝑎, 𝑏] = [0, 2𝜋]. Then the Fourier series (7.1) of 𝑓 converges absolutely. A somewhat stronger result that was proved in [287] reads as follows: Theorem 7.17. Let 1 ≤ 𝑞 < ∞, 𝑞󸀠 = 𝑞/(𝑞 − 1), and 1 ≤ 𝑝 < 2𝑞. Let 𝑓 ∈ Λ𝐵𝑉𝑝 ([0, 2𝜋]) be 2𝜋-periodic. Suppose that ∞

∑

𝜔𝑝+(2−𝑝)𝑞󸀠 (𝑓; 2𝜋/𝑛)1−𝑝/2𝑞 √𝑛𝜆[1, 𝑛]1/2𝑞

𝑛=1

< ∞,

(7.70)

where we have used the shortcut (2.42) for 𝜆[1, 𝑛]. Then the Fourier series (7.1) of 𝑓 con­ verges absolutely. Theorem 7.17 is stronger than Theorem 7.16 because 𝜆[1, 𝑛] ≥ 𝑛𝜆 𝑛, and hence 󸀠

√𝑛𝜆[1, 𝑛]1/2𝑞 ≥ 𝑛(𝑞+1)/2𝑞 𝜆1/2𝑞 = 𝑛1−1/2𝑞 𝜆1/2𝑞 . 𝑛 𝑛 Observe that in case 𝑞 = 1, i.e. 𝑞󸀠 = ∞, the term 𝜔𝑝+(2−𝑝)𝑞󸀠 (𝑓; 2𝜋/𝑛) in (7.70) has to be interpreted as the ordinary modulus of continuity 𝜔∞ (𝑓; 2𝜋/𝑛) defined in (0.97). In particular, for 𝑝 = 𝑞 = 1, condition (7.70) becomes ∞

𝜔∞ (𝑓; 2𝜋/𝑛)1/2 < ∞, √𝑛𝜆[1, 𝑛] 𝑛=1 ∑

(7.71)

which is the condition used by Wang [313]; compare this with (7.22). A parallel result for the general space Λ𝐵𝑉𝜙 ([0, 2𝜋]) of functions of bounded (𝜙, Λ)-variation has been obtained by Schramm and Waterman [287] under the hy­ pothesis that the Young function 𝜙 satisfies the 𝛥 2 -condition (0.21).

7.4 Comments on Chapter 7

| 447

Theorem 7.18. Let Λ = (𝜆 𝑛 )𝑛 be a Waterman sequence, 𝜙 ∈ 𝛥 2 a Young function, 1 ≤ 𝑞 < ∞, 𝑞󸀠 = 𝑞/(𝑞 − 1), and 1 ≤ 𝑝 < 2𝑞. Let 𝑓 ∈ Λ𝐵𝑉𝜙 ([0, 2𝜋]) be 2𝜋-periodic. Suppose that 1/2𝑞 ∞ 𝜔𝑝+(2−𝑝)𝑞󸀠 (𝑓; 2𝜋/𝑛)2𝑞−𝑝 1 )] ∑ < ∞, (7.72) [𝜙−1 ( 𝜆[1, 𝑛] 𝑛=1 √𝑛 where we have again used the shortcut (2.42). Then the Fourier series (7.1) of 𝑓 converges absolutely. Putting 𝜙(𝑡) = 𝑡𝑝 , i.e. Λ𝐵𝑉𝜙 = Λ𝐵𝑉𝑝 in Theorem 7.18, condition (7.72) becomes ∞

∑

𝜔𝑝+(2−𝑝)𝑞󸀠 (𝑓; 2𝜋/𝑛)1/𝑝−1/2𝑞

𝑛=1

√𝑛𝜆[1, 𝑛]1/2𝑝𝑞

< ∞,

(7.73)

which in case 𝑝 > 1, is different from (7.70) since 1−

𝑝 1 1 1 1 = 𝑝( − ) > − . 2𝑞 𝑝 2𝑞 𝑝 2𝑞

Therefore, unfortunately Theorem 7.18 does not contain Theorem 7.17 as a special case. Apart from the moduli of continuity, the modulus of variation (2.52) also turns out to be quite useful in the study of the Fourier series and has been considered by Chan­ turiya in a series of papers [79–84]. For example, the following result is mentioned in [79]; for a proof, see [303]: Theorem 7.19. Let 𝑓 ∈ 𝐶([0, 2𝜋]) be 2𝜋-periodic, and suppose that ∞

𝜈(𝑓)𝑛 < ∞, 2 𝑛=1 𝑛 ∑

(7.74)

where 𝜈(𝑓)𝑛 is defined by (2.52). Then the Fourier series (7.1) of 𝑓 converges uniformly. We point out that Theorem 7.19 implies that if 𝑓 ∈ 𝐶([0, 2𝜋]) ∩ 𝑊𝐵𝑉𝜙 ([0, 2𝜋]) is 2𝜋-periodic and ∞ 1 1 (7.75) ∑ 𝜙−1 ( ) < ∞ , 𝑛 𝑛=1 𝑛 then the Fourier series (7.1) of 𝑓 converges uniformly. To see this, note that 𝑊𝐵𝑉𝜙 ([0, 2𝜋]) ⊆ 𝑉𝜈 ([0, 2𝜋]) for 1 𝜈𝑛 = 𝑛𝜙−1 ( ) , 𝑛 as Proposition 2.36 (or Table 2.2) shows. Therefore, (7.75) implies (7.74), which estab­ lishes the claim. Interestingly, the condition (7.15) (involving the conjugate of 𝜙) re­ duces in the special case 𝜙(𝑡) = 𝑡𝑝 to the condition ∞

1 < ∞, 𝑝/(𝑝−1) 𝑛 𝑛=1 ∑

448 | 7 Some applications which is fulfilled for 1 < 𝑝 < ∞, while the condition (7.75) (involving the inverse of 𝜙) reduces to the condition ∞ 1 ∑ (𝑝+1)/𝑝 < ∞ , 𝑛=1 𝑛 which is fulfilled for 1 ≤ 𝑝 < ∞. So, (7.75) also covers the limit case 𝑝 = 1, while (7.15) does not. If we suppose that the Young function 𝜙 satisfies condition ∞1 (Definition 2.11), which excludes the case 𝑝 = 1 for 𝜙(𝑡) = 𝑡𝑝 , then one may show that (7.15) and (7.75) are equivalent. Moreover, putting 1 𝜉𝑛 := (𝜙∗ )󸀠 ( ) 𝑛

(𝑛 = 1, 2, 3, . . .) ,

Oskolkov [239] has shown that (7.15) is also equivalent to each of the three condi­ tions, namely, 1

∫ log ( 0 ∞

lim 𝜉𝑛 = 0,

𝑛→∞

1 𝜙󸀠 (𝑡)

) 𝑑𝑡 < ∞ ,

∑ (𝜉𝑛 − 𝜉𝑛+1 ) log 𝑛 < ∞

(7.76)

(7.77)

𝑛=1

and ∞

𝜉𝑛 < ∞, 𝑛=1 𝑛 ∑

(7.78)

which for 𝜙(𝑡) = 𝑡𝑝 , all hold in case 1 < 𝑝 < ∞. In fact, the proof of the equivalence of (7.15) and (7.78) relies on the estimate 𝜉 𝜉2𝑛 1 ≤ 𝜙∗ ( ) ≤ 𝑛 2𝑛 𝑛 𝑛 which follows from the convexity of 𝜙. Being the derivative (a.e.) of a convex function, 𝜙󸀠 is monotonically increasing, and so the set 𝑀𝑛 := {𝑡 : 0 ≤ 𝑡 < ∞,

1 1 ≤ 𝜙󸀠 (𝑡) < } 𝑛+1 𝑛

satisfies (𝜉𝑛+1 , 𝜉𝑛 ) ⊆ 𝑀𝑛 ⊆ [𝜉𝑛+1 , 𝜉𝑛 ], and hence has Lebesgue measure 𝜆(𝑀𝑛 ) = 𝜉𝑛 −𝜉𝑛+1 . Consequently, (𝜉𝑛 − 𝜉𝑛+1 ) log 𝑛 ≤ ∫ log ( 𝑀𝑛

1 ) 𝑑𝑡 ≤ (𝜉𝑛 − 𝜉𝑛+1 ) log(𝑛 + 1) , 𝜙(𝑡)

which establishes the equivalence of (7.76) and (7.77). Finally, the estimate 𝑡 󸀠 𝑡 𝜙 ( ) ≤ 𝜙(𝑡) ≤ 𝑡𝜙󸀠 (𝑡) , 2 2

7.4 Comments on Chapter 7

|

449

together with the convexity of 𝜙 and the condition 𝜙(0) = 0, implies the equivalence of (7.15) and (7.76). In Proposition 2.85, we have shown that Λ𝐵𝑉𝜙 ([0, 2𝜋]) ⊆ 𝑉𝜈 ([0, 2𝜋]) for 𝜈𝑛 = 𝑛𝜙−1 (

1 ). 𝜆[1, 𝑛]

So, as a consequence of Theorem 7.19, we obtain the following Corollary 7.20. Let 𝑓 ∈ 𝐶([0, 2𝜋]) be 2𝜋-periodic, and suppose that ∞

1 −1 1 ) < ∞, 𝜙 ( 𝑛 𝜆[1, 𝑛] 𝑛=1 ∑

(7.79)

where 𝜙 is some Young function, Λ = (𝜆 𝑛 )𝑛 is a Waterman sequence, and 𝜆[1, 𝑛] is defined by (2.42). Then the Fourier series (7.1) of 𝑓 converges uniformly. In connection with the convergence criteria we just discussed, the following interest­ ing result involving the Banach indicatrix (0.106) of a continuous function is proved in [239, Theorem 3]: Proposition 7.21. Let 𝜏 : [0, ∞) → [0, ∞) be an increasing function such that 𝜏(0) = 0, 𝜏(𝑡) > 0 for 𝑡 > 0, and 1

∫ log ( 0

1 ) 𝑑𝑡 = ∞ . 𝜏(𝑡)

(7.80)

Then there exists a 2𝜋-periodic function 𝑓 ∈ 𝐶([0, 2𝜋]) whose Banach indicatrix (0.106) satisfies 1 (𝑥 > 0) 𝐼𝑓 (𝑥) ≤ 𝜏(𝑥) and whose Fourier series (7.1) diverges at some point. Proposition 7.21 shows that the Goffman–Waterman criterion (7.15) is, in a certain sense, sharp. To see this, observe that under the hypotheses of Proposition 7.21, we have 𝑓 ∈ 𝑊𝐵𝑉𝜙 ([0, 2𝜋]), where¹⁰ 𝑡

𝜙(𝑡) := ∫ 𝜏(𝑠) 𝑑𝑠

(𝑡 ≥ 0) .

0

Condition (7.80) then means that (7.76) fails, and thus also (7.15). Let us mention another result of Chanturiya [79] which is formulated in terms of a combination of both the modulus of continuity (0.97) and the modulus of variation (2.52).

10 The convexity of 𝜙 follows from the fact that 𝜏 is increasing.

450 | 7 Some applications Theorem 7.22. Suppose that there exists some 𝛼 ∈ (0, 1] such that either 𝜈(𝑓)𝑛 = 𝑂 ( and 𝜔∞ (𝑓; 𝛿) = 𝑂 ( or 𝜈(𝑓)𝑛 = 𝑂 (

𝑛 ) (log 𝑛)(log log 𝑛)𝛼

1 ) exp(𝑜(log log(1/𝛿))𝛼 ) , log(1/𝛿)

𝑛 ) (log 𝑛)(log log 𝑛)(log log log 𝑛)𝛼

and

𝛼

𝜔∞ (𝑓; 𝛿) = 𝑂 (log(1/𝛿)− exp(𝑜(log log log(1/𝛿)) ) ) . Then the Fourier series (7.1) of 𝑓 converges uniformly. Apart from convergence criteria for the Fourier series (7.1), summability results for the Fourier coefficients (7.2) and (7.3) are also of interest. For example, in [317], the author shows that for 𝑓 ∈ Λ𝐵𝑉, the Fourier coefficients of 𝑓 satisfy 𝛼𝑛 (𝑓), 𝛽𝑛 (𝑓) = 𝑂 (

1 ) 𝑛𝜆 𝑛

(𝑛 → ∞) .

Finally, let us briefly discuss two results on the rate of convergence of the partial Fourier sums (7.4), the first in the Wiener–Young space 𝑊𝐵𝑉𝜙 ([0, 2𝜋]), the second in the special Waterman space Λ 𝑞 𝐵𝑉([0, 2𝜋]). Proposition 7.23. Let 𝜙 be a Young function and 𝑓 ∈ 𝐶([0, 2𝜋]) ∩ 𝑊𝐵𝑉𝜙 ([0, 2𝜋]). Then 𝜔∞ (𝑓;𝜋/𝑛)

‖𝑓 − 𝑠𝑛 (⋅; 𝑓)‖𝑊𝐵𝑉𝜙 ≤ 𝑐

∫ 0

log

Var𝑊 𝜙 (𝑓; [0, 2𝜋]) 𝜙(𝑡)

𝑑𝑡 ,

where 𝜔(𝑓; 𝛿) denotes the modulus of continuity (0.97) of 𝑓, and Var𝑊 𝜙 (𝑓; [𝑎, 𝑏]) the Wiener–Young variation (2.2) of 𝑓. Proposition 7.23 has been proved by Oskolkov [239]; in the special case 𝜙(𝑡) = 𝑡 (i.e. 𝑊𝐵𝑉𝜙 = 𝐵𝑉), the result goes back to Stechkin [298]. Convergence results involving the average sums (7.7) may be found in [5, 23]. The following proposition provides a pointwise estimate for the rate of conver­ gence of (7.4) to 𝑓(𝑥). Proposition 7.24. Let 0 < 𝑞 < 1, and let 𝑓 ∈ 𝐶([0, 2𝜋]) ∩ Λ 𝑞 𝐵𝑉([0, 2𝜋]). Then |𝑓(𝑥) − 𝑠𝑛 (𝑥; 𝑓)| ≤

(2 − 𝑞)(1 + 2/𝜋) 𝑛 VarΛ 𝑞 (𝜑𝑥 ; [0, 𝜋/𝑘]) ∑ , (𝑛 + 1)1−𝑞 𝑘𝑞 𝑘=1

where 𝜑𝑥 denotes the auxiliary function (7.10), and VarΛ 𝑞 (𝑓; [𝑎, 𝑏]) the Waterman varia­ tion of 𝑓 given in Definition 2.29.

7.4 Comments on Chapter 7

| 451

Proposition 7.24 is due to Bojanić and Waterman [51], in the special case 𝑞 = 0 (i.e. Λ 𝑞 𝐵𝑉 = 𝐵𝑉) this is an earlier result of Bojanić [50]. For a more general result (classes of type Λ𝐵𝑉 which can be closer to 𝐻𝐵𝑉), see Waterman [318]. A completely different approach to the convergence of Fourier series based on the Korenblum variation can be found in [164]. As Exercise 2.40 shows, a function 𝑓 ∈ 𝜅𝐵𝑉([𝑎, 𝑏]) ∩ 𝐶([𝑎, 𝑏]) is not necessarily of vanishing 𝜅-variation. In [164, Theorem 3], it is shown that if 𝑓 ∈ 𝜅𝐵𝑉([0, 2𝜋]), where 𝜅(𝑡) = 1/(1 − 12 log 𝑡), and the function 𝜑𝑥 defined in (7.10) has vanishing 𝜅-variation at 𝑡0 , then 𝑠𝑛(𝑡0 ; 𝑓) → 𝑓(𝑡0 ) as 𝑛 → ∞, with 𝑠𝑛(⋅; 𝑓) defined by (7.4). To conclude, let us make some comments on Section 7.3. Solutions of nonlinear integral equations in spaces of type 𝑊𝐵𝑉𝑝 or Λ𝐵𝑉 have been studied by Bugajew­ ska, Bugajewski, and others [67–72]. One could ask why we did not give more explicit conditions on the functions ℎ which guarantee that the superposition operator 𝑆ℎ in (7.45) maps the space 𝐵𝑉𝑝 or the space Λ𝐵𝑉 into itself. The reason is simply that con­ ditions which are both necessary and sufficient are not known, and conditions which are only necessary (like Theorem 6.10) or only sufficient (like Theorem 6.11) are rather technical. The “natural” requirement that ℎ(𝑥, ⋅) is locally Lipschitz, uniformly in 𝑥, and ℎ(⋅, 𝑢) belongs to 𝐵𝑉𝑝 , respectively Λ𝐵𝑉, uniformly in 𝑢, is not sufficient, as the surprising Example 6.8 shows. In the last part of Section 7.3, we have studied the singular nonlinear equation (7.61) in Hölder spaces. Such equations are considered in so-called generalized Hölder spaces by Babaev [37] and Mukhtarov [234], see also Chapter 5 of the book [138]. Exis­ tence and uniqueness results for solutions of (7.61) are closely related to existence and uniqueness results for solutions of initial or boundary value problems for ordinary dif­ ferential equations. In view of similar applications to partial differential equations, it is useful to formulate analogous results for functions which are defined on a domain 𝛺 ⊆ ℝ𝑛 (not necessarily bounded). To this end, one has to work in the space 𝐿𝑖𝑝𝛼 (𝛺) of all bounded continuous functions 𝑓 : 𝛺 → ℝ with norm ‖𝑓‖𝐿𝑖𝑝𝛼 := ‖𝑓‖∞ + sup 𝑥=𝑦 ̸

|𝑓(𝑥) − 𝑓(𝑦)| , |𝑥 − 𝑦|𝛼

(7.81)

where ‖𝑓‖∞ now denotes the supremum of |𝑓(𝑥)| on 𝛺. Here, we have the following analogue to Theorem 5.51: Theorem 7.25. Suppose that the function ℎ : ℝ → ℝ is differentiable on ℝ and satisfies condition (5.14). Then the composition operator 𝐶ℎ 𝑓 = ℎ∘𝑓 maps the space 𝑋 = 𝐿𝑖𝑝𝛼 (𝛺) into itself and satisfies the local Lipschitz condition (5.76) in the norm (7.81). The proof consists of an evident modification of the arguments used in the proof of Theorem 5.51 in Chapter 5.

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List of functions In many places within the first chapters of this book, we have shown that the inclusion 𝑋 ⊂ 𝑌 between two function spaces 𝑋 and 𝑌 is strict by constructing a function 𝑓 ∈ 𝑌\𝑋. For the reader’s ease, we have provided some of these functions here and indicate where to find them. – For 1 ≤ 𝑝 < ∞, a function 𝑓 ∈ 𝐿 𝑝 ([0, 1]) \ (∪𝑞>𝑝 𝐿 𝑞 ([0, 1])): Example 0.11. – For 1 < 𝑝 ≤ ∞, a function 𝑓 ∈ (∩𝑞 1 and 𝑝 − 1 ≥ 𝑝𝑞, a function 𝑓 ∈ 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏]) \ (∪𝑝−1≥𝑝𝑞 Λ 𝑞 𝐵𝑉([𝑎, 𝑏])): Exercise 2.49. A function 𝑓 ∈ 𝐻𝐵𝑉([𝑎, 𝑏]) \ 𝐵𝑉([𝑎, 𝑏]): Example 2.30. A function 𝑓 ∈ 𝐻𝐵𝑉([𝑎, 𝑏]) \ (∪𝑝>1 𝑊𝐵𝑉𝑝 ([𝑎, 𝑏])): Exercise 2.7. For 1 ≤ 𝑝 < 𝑞, a function 𝑓 ∈ 𝑅𝐵𝑉𝑝 ([𝑎, 𝑏]) \ 𝑅𝐵𝑉𝑞 ([𝑎, 𝑏]): Exercise 2.1. For 0 < 𝑞 < 1, a function 𝑓 ∈ 𝐿𝑖𝑝1−𝑞 ([𝑎, 𝑏]) \ Λ 𝑞 𝐵𝑉([𝑎, 𝑏]): Exercise 2.83. For 0 < 𝑞 < 1, a function 𝑓 ∈ V𝜈𝑞 ([𝑎, 𝑏]) \ 𝑊𝐵𝑉1/(1−𝑞) ([𝑎, 𝑏]): Example 2.40. For 0 < 𝑞 < 𝑟 < 1, a function 𝑓 ∈ Λ𝑐𝑟 𝐵𝑉([𝑎, 𝑏]) \ V𝜈𝑞 ([𝑎, 𝑏]): Example 2.41. A function 𝑓 ∈ (∩𝛼